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Experimental study of the current filamentation instability

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Experimental study of the current filamentation instability

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Content EXPERIMENTAL STUDY OF THE CURRENT FILAMENTATION INSTABILITY by Brian Allen A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) December 2012 Copyright 2012 Brian Allen ii Dedication To my wife and parents. iii Acknowledgments This quest started before graduation from my undergraduate edu- cation, but the first time it was communicated was when I told my father that I would finish my PhD immediately after graduation. Little did I, or he, know that it would be a decade later and in a completely different dis- cipline. One way I view life is as freeway speeding one towards their destina- tion. There are many off ramps, interchanges and perhaps rest stops along the way. A quest also typically includes meeting many people along the way, a few played important roles and others provided memorable events. My quest would never have started without my first electromagnet- ics course back in 1999 with Tom Katsouleas. A teacher or professor who is excited about a topic and able to inspire students is truly rare in my ex- perience and Tom is blessed with this combination. A few conversations outside of class about plasma physics and plasma based accelerators left a lasting impression and fostered an interest in the subject. My quest was almost derailed before it began when two months after I started Tom accepted the opportunity to be the Dean of Duke Engineering school. He left the group in Patric Muggli’s hands, whom I am not sure iv I had met at that point and definitely had not had a serious conversation with. Over the next four years I gained a deep respect for Patric, admire his true passion for science and especially appreciated his belief that things should be done right or not done at all. His willingness and ability to share his knowledge is unparalleled. Together we traveled to three continents and made numerous trips to Brookhaven National Laboratory which spanned several months of our lives and I can say that I enjoyed and learned from every one of those experiences. Vitaly Yakimenko is one of those characters you meet in life who you will think about from time to time and in hindsight will appear larger than life. Vitaly is one of great knowledge, experience, interesting ideas and boundless confidence. His never say no attitude and leadership of the ATF played a significant role in the success of the experiment. I am greatly appreciative to the ATF staff without which the ex- periment would never have succeeded. Special thanks to Karl Kusche for his ideas, conversation and tireless work including early mornings and late nights to ensure the success of the experiment. Marcus Babzien for sharing a small portion of his vast optics and laser knowledge with me and spending time to help me understand topics and concepts. Mikhail Fedurin for his expert command of the electron beam and his ability to make the long hours at the ATF seem short. Finally to the remainder of the ATF staff. Growth academically is greatly improved and accelerated with/by v others that challenge you and ask questions you might not have considered or tried to avoid. These traits are especially true with Yun Fang for helping me truly learn and understand ideas in both of our research areas. She also provided motivation, productive and stimulating conversations and a good traveling companion. Thanks to all of my former group mates for great discussions and taking their time to discuss research and science: Yi, Bing, Xiaodong, Reza, Themos, Xiaoing and Stephen. No endeavor such as a PhD can ever be undertaken without great family and friends. My parents provided the foundation for anything that I have ever accomplished and have always supported and believed in me despite voicing concern at some points. In 2006 my own family started when I married my wife, SungIn. She has always encouraged me to pursue my passions and has been there for me whenever I needed encouragement. She is always a willing distraction in my life. Steve Suzuki for being with me on my adventures since high school. You helped take my mind off of my research, explore restaurants and go on random trips. Keith Wall kept my mind sharp and always challenged my ideas. Thanks to other friends and their families: Tushaar and Kristin, Kim and Brooks, Cat, Megan, Gigi and Marlon, Jess and Mike and all the rest of you I might have forgotten. vi Table of Contents Dedication ii Acknowledgments iii List of Tables ix List of Figures x Abstract xvi Chapter 1. Introduction 1 1.1 Plasma and Plasma Instabilities . . . . . . . . . . . . . . . . . . 1 1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 PWFA - Limitation to maximum accelerating gradient . 4 1.2.2 Afterglow from Gamma-ray Bursts . . . . . . . . . . . . 5 1.2.3 Fast Igniter - Inertial Confinement Fusion . . . . . . . . . 7 1.3 Thesis Goals and Structure . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Goals of Thesis . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Structure of Thesis . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 2. Theoretical/Technical Review of CFI 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Plasma Focusing and Return Currents . . . . . . . . . . . . . . 14 2.3 Plasma Instabilities: Weibel, Two-Stream and Current Filamentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.1 Weibel Instability . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.2 Two-Stream Instability . . . . . . . . . . . . . . . . . . . . 20 2.3.3 Current Filamentation Instability . . . . . . . . . . . . . . 21 vii 2.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . 28 Chapter 3. Simulation Study of CFI 30 3.1 Particle-in-Cell Codes for Plasma Simulations . . . . . . . . . . 31 3.2 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Simulations Results . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Simulations with “Nominal” Parameters . . . . . . . . . 36 3.3.2 Function of Beam Density . . . . . . . . . . . . . . . . . . 40 3.3.3 Function of Plasma Density . . . . . . . . . . . . . . . . . 43 3.3.4 Function of Beam Emittance . . . . . . . . . . . . . . . . 48 3.4 Method for Seeding of CFI Through the Initial Beam Profile . . 51 3.5 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 56 Chapter 4. Experimental Setup 58 4.1 Experimental Components and Setup . . . . . . . . . . . . . . . 58 4.1.1 ATF Facility and 60 MeV Linear Accelerator . . . . . . . 58 4.1.2 Plasma Source and Density Diagnostic . . . . . . . . . . 60 4.1.3 Direct Imaging System . . . . . . . . . . . . . . . . . . . . 61 4.1.4 Imaging System Design and Components . . . . . . . . 66 4.2 Resolution and Magnification of Imaging System . . . . . . . . 69 4.3 Summary/Conclusion . . . . . . . . . . . . . . . . . . . . . . . 71 Chapter 5. CFI Experimental Results 72 5.1 Previous Experimensts . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 77 5.3.1 Transverse Imaging of Beam and Multiple Filaments . . 77 5.3.2 Beam-Plasma Interaction Regime . . . . . . . . . . . . . 78 5.3.3 Transverse Filament Size Scaling . . . . . . . . . . . . . . 80 5.3.4 Instability Growth as a Function of Beam Density . . . . 81 5.4 Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 84 Chapter 6. Conclusions 86 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 viii 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Bibliography 92 Appendix A. Specification/Data sheet for Princeton Instruments EMCCD Camera - ProEM1024B 99 Appendix B. Specification/Data sheet for Ealing 25-0506-000 micro- scope objective 105 Appendix C. CFI Alignment Process 107 ix List of Tables Table 3.1 ATF over-compressed electron beam parameters. . . . . . . . 34 Table 3.2 “Nominal” simulation beam and plasma parameters. . . . . 36 Table 3.3 Summary of growth rate, regime and average rms filament size as the plasma density increases for “nominal” beam parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Table 3.4 Growth rate and average transverse filament rms sizeL p = 4 cm for four normalized emittances and otherwise “nomi- nal” beam and plasma parameters. . . . . . . . . . . . . . . . 51 Table 3.5 Growth rate and growth length of CFI with 10m wide cross for increased intersected particle momenta (see Fig. 3.16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Table 4.1 Growth in filament size due to scattering from Si window for filament size =10:6m. Radiation length X o : Si - 21:82 g=cm 2 [15] . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Table 4.2 Growth in filament size due to scattering from Au layer for filament size =10:6m. Radiation length X o : Au - 6:46 g=cm 2 [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Table 5.1 Beam parameters, from discrete scan, for the high and low charge cases, Fig. 5.3, and corresponding growth rate, growth length and number of e-foldings overL p = 2 cm. . . 83 x List of Figures Figure 1.1 Image of Collapsar model. Accretion disk with material being ejected from the rotational axis [1]. . . . . . . . . . . . 6 Figure 1.2 Inertial confinement fusion sequence [2]. . . . . . . . . . . . 9 Figure 1.3 Fast-Ignition ICF with Cone-in-shell concept with gold cone imbedded into a fuel pellet. Inset is X-ray image showing imploded core at tip of cone [32]. . . . . . . . . . . 10 Figure 2.1 Electric field and wave vector for current filamentation, two-stream and Weibel instabilities. Z-axis is the propa- gation axis for the beam for the current filamentation and two-stream instabilities and the low temperature axis for the Weibel instability. . . . . . . . . . . . . . . . . . . . . . . 19 Figure 2.2 Magnetic energy and beam density evolution, simulations from chapter 3 for “nominal” case, section 3.3.1. Three steps in CFI evolution are denoted A) linear growth, B) saturation and C) filament coalescence. Color map is the same for all beam density images. . . . . . . . . . . . . . . . 22 Figure 3.1 Particle-in-cell computational cycle. . . . . . . . . . . . . . . 33 Figure 3.2 Integrated beam density showing the evolution of CFI for “nominal” beam in a uniform plasma density n e = 5 10 17 cm 3 . Note that L p = 0:2 cm can be considered the same asL p = 0:0 cm as CFI has not yet developed. Color map is the same for all images. . . . . . . . . . . . . . . . . 38 xi Figure 3.3 Integrated beam density of the evolution of the beam/instability for “nominal” beam parameters in plasma density of n e = 5 10 17 cm 3 . Letters denote image corresponding to those in Fig. 3.2. Note thatL p = 0:2 cm can be considered the same asL p = 0:0 cm as CFI has not yet developed. . . . 39 Figure 3.4 Magnetic energy in simulation box versus plasma length for “nominal” case. Vertical lines and letters correspond to those of Fig. 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . 40 Figure 3.5 Integrated x and y magnetic fields (B x +B y ) for the “nom- inal” beam parameters in a uniform plasma densityn e = 5 10 17 cm 3 . Color map is the same for all images. . . . . . 41 Figure 3.6 Magnetic energy over 4 cm of uniform plasma den- sity n e = 5 10 17 cm 3 for three beam charges Q = 56; 112 and 225 pC. Dashed vertical lines represent ap- proximate saturation length. Saturation is not reached for Q = 56 pC. Solid lines are fits to the exponential growth and the slope represents the growth rate. . . . . . . . . . . . 42 Figure 3.7 Integrated transverse beam density at L p = 2:5 cm with plasma density n e = 5 10 17 cm 3 and otherwise “nomi- nal” beam parameters for three beam charges. Color map is the same for all three images. . . . . . . . . . . . . . . . . 43 Figure 3.8 Growth rate from theory (red line) and simulations (blue points) for three beam charges. Solid blue line is the power fit to the data points. . . . . . . . . . . . . . . . . . . . . . . . 44 Figure 3.9 Transverse rms filament sizes atL p = 4 cm for plasma skin depthsc=! pe (n e ) = 5:3m (1:0010 18 cm 3 ); 6:3m (7:07; 7:5 (5:00 10 17 cm 3 ); and 8:9m (3:54 10 17 cm 3 ). Red line is fit of data points and black line shows linear scaling. . . . . . . . 45 xii Figure 3.10 Peak beam density for plasma skin depth (plasma density) c=! pe (n e ) = 5:3 (10:00); 6:3 (7:07); 7:5 (5:00); and 8:9m (3:54 10 17 cm 3 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 3.11 Magnetic energy for four plasma densities and “nominal” beam parameters. Dashed vertical lines represent approx- imate CFI saturation length. . . . . . . . . . . . . . . . . . . 47 Figure 3.12 Filament sizes, rms, for plasma densities (skin depths) in Fig. 3.11 and three lower plasma densities c=! p = 37:5 (2:010 16 cm 3 ); 53:0m (110 16 cm 3 ) and 75:0m (5:0 10 15 cm 3 ). Solid line shows ideal scaling of CFI. . . . . . . . 48 Figure 3.13 Magnetic energy overL p = 4:0 cm of uniform plasma den- sityn e = 5 10 17 cm 3 for four normalized emittances and otherwise “nominal” beam parameters. Dashed vertical lines represent approximate saturation lengths. Solid lines indicate the growth rate. . . . . . . . . . . . . . . . . . . . . . 50 Figure 3.14 Integrated transverse beam density atL p = 4 cm for “nom- inal” beam and plasma parameters for normalized emit- tances of Fig. 3.13. There are four very weak filaments for " Nx;y = 8 mmmrad and only a single feature for " Nx;y = 12 mmmrad. Color maps are the same for all images. . . . 50 Figure 3.15 Transverse summed beam density for propagation dis- tances of L p = 0:1; 2:0 and 3:1 cm in a uniform plasma of density n e = 2:5 10 17 cm 3 for a beam that is a) un-modified and for increased particle momentum (P x and P y ) of particles in 10m width cross increased by b) p 10x, c) 10x and d) particles removed. Color maps are the same for all images. . . . . . . . . . . . . . . . . . . . . . . . 54 xiii Figure 3.16 Magnetic energy in the simulation box as a function of propagation distance in a uniform plasma density n e = 2:5 10 17 cm 3 for the unmodified beam (red symbols) and particles momenta along 10m wide cross pattern increased by p 10x (blue symbols), 10x (green sym- bols) and infinity - particles removed - (purple symbols). Slope of fitted solid lines represent growth rate. Dashed vertical lines represent approximate saturation distance. . . 55 Figure 4.1 Experimental setup: electron bunch from a 60MeV linac is focused to its waist at the entrance to the capillary. The plasma is generated by puffing H 2 gas at 110 Torr into a 1 mm diameter by 2 cm long capillary with an applied volt- age of 15 kV. OTR is generated from the bunch/filament electrons and the Au coating. The resulting light is collected and magnified by a microscope objective and relayed with turning mirrors out of the vacuum chamber then focused onto an EMCCD camera. The camera, lens and closest 3” turning mirror are on a plane lower than the other components, Fig. 4.6. . . . . . . . . . . . . . . . . . 59 Figure 4.2 Layout of BNL-ATF experimental hall with the three beam lines. The experiment interaction point is the center of the Compton chamber on beam line one [3]. . . . . . . . . . . . 60 Figure 4.3 (a) Capillary assembly from downstream side, small hole in window holder is capillary exit, 1 mm diameter and (b) capillary assembly in experimental configuration. . . . . 61 Figure 4.4 Transition radiation generated (forward and backward) from an electron propagating from medium 1, " 1 = ", to medium 2 - vacuum," 2 = 1. . . . . . . . . . . . . . . . . . . . 62 xiv Figure 4.5 In-vacuum imaging system components: microscope ob- jective and 1” turning mirror. Capillary exit, capillary not shown, is 25 mm upstream (to the left) of microscope ob- jective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Figure 4.6 Out of vacuum imaging system configuration: 3” turn- ing mirrors, camera lens and EMCCD camera. The 135 mm camera lens is not shown. Note that the camera is 12” below the beam path and that lead shielding (not shown) surrounded the camera. . . . . . . . . . . . . . . . . . . . . . 69 Figure 4.7 Longitudinal (direct) filament imaging for the imaging system setup on beam line 1, but with a USAF 1951 resolution test target placed at the location of the Si/Au window on the downstream side of the capillary. . . . . . . 70 Figure 5.1 OTR images of the bunch at the capillary exit with arrows indicating filaments. Bunch parameters are 0x 80m, 0y 50m and charge Q ' 1:0nC. a) without plasma and b) n e = 1:6 10 16 cm 3 (c=! pe = 41:6m), c) and d) n e = 1:2 10 17 cm 3 (c=! pe = 15:4m) and e) and f)n e = 1:9 10 17 cm 3 (c=! pe = 12:3m). X-rays are seen in the images and are a few pixels in size. Color tables are the same for b) through f). . . . . . . . . . . . . . . . . . . . . . . 79 Figure 5.2 Number of filaments observed in single events as a func- tion of the CFI parameterk p 0y . For this measurement the charge Q' 1:0 nC and 0x 80m, 0y 50m. Sim- ilar events lead to only single filaments for k p 0y < 2:2 and multiple (one to five) filaments for k p 0y > 2:2. For k p 0y > 4:5 only one and two filaments are seen and could be due to merging of filaments. In general single points represent multiple events at the samek p 0y . . . . . . . . . . 80 xv Figure 5.3 Transverse filament size (RMS): red markers high charge scan - Q ' 1:0 nC and 0x = 81m, 0y = 53m. Blue markers low charge scan -Q' 0:54 nC and 0x ' 89m, 0y ' 45m. Both scans consist of six different plasma densities with each ten events recorded. The solid line rep- resents exact correlation between filament size and plasma skin depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Figure 5.4 Transverse filament size (RMS): red markers high charge scan - Q ' 1:0 nC and 0x = 80m, 0y = 53m. Blue markers low charge scan -Q' 0:54 nC and 0x ' 91m, 0y ' 48m. Both scans consist of 56 different plasma den- sities each with one event recorded. The solid line repre- sents exact correlation between filament size and plasma skin depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Figure 6.1 Proposed transverse and longitudinal diagnostics setup. . . 89 xvi Abstract Plasma instabilities can produce strong anisotropies, accelerate particles to high energies and generate large electric and magnetic fields. The Current Filamentation Instability, CFI, is of central importance for the propagation of relativistic electron beams in plasmas. CFI has potential relevance to astrophysics (afterglow of gamma ray bursts), inertial confine- ment fusion (energy transport in the fast-ignitor concept) and beam-driven plasma-based accelerators (placing an upper limit on the plasma density and accelerating gradient). The work in this dissertation combines a review of theory, particle-in- cell simulations and design and execution of an experiment that resulted in the first conclusive observation of the current filamentation instability. Cur- rent theory of CFI is reviewed and discussed. Simulations, with a particle- in-cell code, were conducted to validate theory for the finite size bunch and plasma used in the work and provide insight for the experiment. The design of a high-magnification imaging system to observe the result of the instabil- ity and the setup of the experiment are detailed. Finally, the experimental results are presented and discussed. The experiment was conducted at the Accelerator Test Facility at xvii Brookhaven National Laboratory with the 60MeV e bunch and 2cm long plasma with variable density. The experiment included the systematic study and characterization of the instability as a function of the beam charge and plasma density, or ratio of bunch transverse size to plasma skin depthc=! p e or 1=k pe . The transverse beam profile and size ( r ) is measured directly at the plasma exit using optical transition radiation from a thin gold coated silicon window. Experimental results show the transition from plasma focusing (k p r << 1) to CFI (k p r >> 1) is characterized by the appearance of multiple (1 5) beam filaments and the linear scaling of the transverse filaments size with the plasma skin depth. Suppression of the instability is demonstrated when lowering the growth rate of the instability by reducing the beam charge. The experimental results are in excellent agreement with theory and simulations. 1 Chapter 1 Introduction 1.1 Plasma and Plasma Instabilities A plasma is a distinct phase of matter, often referred to as the fourth state of matter, which has been described as a “quasineutral gas composed of charged and neutral particles and exhibits collective behavior” [12]. It is quasineutral because the number of positive and negative charged particles is very nearly equal. A plasma exhibits collective behavior in that particles act together in a coordinated way. In this thesis our study is confined to cold, uniform and un-magnetized plasmas. A beam is defined to be a finite transverse size stream or pulse of charged particles and is not a plasma because it is not quasineutral. An ideal plasma is one that is in perfect thermodynamic equi- librium with no free energy. Perfect thermal equilibrium is defined to be when the density, electric fields and magnetic fields are uniform and the particles have a maxwellian velocity distribution. When a plasma is not ideal free energy is available and can cause waves to be self excited resulting in an unstable equilibrium. An instability is always a motion which decreases the systems free energy and brings the plasma closer to 2 a perfect thermodynamic equilibrium. Plasma instabilities can produce strong anisotropies, accelerate particles to high energies and generate large electric and magnetic fields. Plasma instabilities can be classified by the source of free energy [12]: 1. Streaming: Different drift velocities resulting from a beam of particles propagating or a current driven through a plasma. The resulting drift energy excites waves and oscillation energy is gained. 2. Raleigh-Taylor: Non-uniformity in plasma density due to a gradient or sharp boundary and the potential energy is released due to an ex- ternal non-electromagnetic force. 3. Universal: Expansion energy, whenever a plasma is confined it is not in thermodynamic equilibrium. 4. Kinetic: Non-maxwellian velocity distributions, deviation from ther- modynamic equilibrium. The field of beam-plasma instabilities is a topic of great research in- terest. A beam propagating in a plasma generates an equal and opposite return current in the plasma and when the return current passes through the beam it is susceptible to beam-plasma instabilities. The return current passes through the beam when the transverse beam size is greater than the plasma collisionless skin depth. The plasma skin depth is an example of the 3 collective behavior of a plasma and is the characteristic length over which a cold plasma can shield charge and current [12]. The plasma skin depth is in- versely proportional to the plasma density, i.e. a high plasma density corre- sponds to a small skin depth. The current filamentation instability, CFI, is a particular beam-plasma instability for relativistic beams and belongs to the streaming category of instabilities. It is an electromagnetic instability with a purely imaginary frequency, similar to the Weibel instability (discussed in chapter two) [54] and characterized by a wave number perpendicular to the beam propagation direction. CFI is seeded by non-uniformities in the beam or plasma transverse current profiles. These non-uniformities lead to unequal and opposite current densities and to the development of local magnetic fields. These current densities repel each other, locally enhancing magnetic fields and exerting a force on the opposite current. This repulsion increases the non-uniformities in the distribution of the current densities and a positive feedback loop is established leading to breakup of the beam into transverse high current density filaments. The CFI has relevance to ad- vanced particle acceleration concepts (plasma based accelerators, PWFA), future energy sources (inertial confinement fusion) and understanding as- trophysical events (afterglow from gamma ray bursts). While CFI is impor- tant very little experimental work has been conducted and no studies of the instability have been reported, this represents the originality of the work. 4 1.2 Motivation 1.2.1 PWFA - Limitation to maximum accelerating gradient Plasma based particle accelerators can be driven by charged parti- cle bunches, PWFA, or laser pulses, LWFA [52]. In the PWFA concept a charged particle bunch, composed of electrons, positrons, protons or ions, propagates in a neutral plasma [13]. The space charge field of the bunch dis- places the plasma electrons and drives a plasma wave, called a wake. The maximum sustainable electric field, or wake, is defined as the cold wave- breaking limit [46] and is proportional to the plasma density. A high plasma density corresponds to a larger acceleration gradient. For an efficient PWFA scheme the charge profile of the incoming beam must be maintained and avoiding the CFI regime which would result in the breakup of the beam is critical. To avoid the CFI regime the plasma skin depth must be larger than the focused transverse beam size. Thus the CFI places a limit on the max- imum plasma density and hence the maximum accelerating gradient for a given focused beam size. To this point CFI has not posed an issue to PWFA experiments at SLAC [7, 8, 22, 42] and BNL-ATF [26, 41, 44]. However, this is now impor- tant for the proton-driven PWFA experiments in the long bunch regime that are in preparation at CERN [43, 56]. To avoid the development of CFI in the bunch the plasma density will be limited, lowering the maximum achievable accelerating gradient. 5 Further study of CFI will determine whether these requirements can be relaxed. 1.2.2 Afterglow from Gamma-ray Bursts Gamma-ray bursts, GRBs, are some of the brightest events in the uni- verse, hundreds of times brighter than a supernova and as bright as a mil- lion trillion suns [31]. These bursts of gamma-ray radiation last from a few milliseconds to hundreds of seconds and are typically followed by longer wavelength radiation emission lasting from hours to weeks. This longer wavelength radiation is termed the ”afterglow” [37, 48]. The phenomena responsible for GRBs and their subsequent afterglow are largely unknown. One set of theories which provides a mechanism for these events is the col- lapsar model, generation of relativistic dense jets of material, and the rela- tivistic fireball shock model, for creation of the GRB and afterglow. When a star depletes its nuclear fuel and cannot support its bulk it collapses into a black hole and expels most of its mass into the interstellar medium, this is called a supernova. In the collapsar model a black hole begins to pull in nearby interstellar material and an accretion disk forms where the inner portion of the disk rotates more rapidly than the outer [34]. The accretion disk is a rotating fluid generating magnetic fields, the mag- netic field lines twist and create a jet of material, consisting of electrons, 6 protons and positrons, exploding out perpendicularly from the disks rota- tional axis at the speed of light, Figure 1.1. Figure 1.1: Image of Collapsar model. Accretion disk with material being ejected from the rotational axis [1]. In the relativistic fireball shock model this relativistic jet of material behaves like a shockwave, and as it propagates outward it sweeps up matter in its path, hence called a fireball [11]. Within the fireball the temperature, pressure and density vary resulting in internal shock waves moving back and forth, the faster moving masses overtake slower masses. Gamma-rays are produced as a result of the collisions of these masses. The longer wavelength, compared to gamma rays, signature of after- 7 glows from GRBs is similar to synchrotron radiation, relativistic electrons (or particles) gyrating in high intensity magnetic fields emit radiation [16, 47]. Synchrotron radiation models are able to reproduce the spectra of some measured observed afterglows from GRBs, however the radiation signature is often different and could be produced by particles trapped in random or tangled magnetic fields and is called ”jitter” radiation [35, 55]. The afterglow occurs when the fireball, and the subsequent colli- sionless shocks, propagate through the interstellar medium, a low density plasma. These shocks impart anisotropies into the interstellar medium and are subject to the Weibel instability [36]. Another way of looking at the afterglow is the ejected material, the fireball, plows through the interstellar medium, which would be similar to the current filamentation instability. This second explanation could be seen as a beam of ions, electrons and positrons, propagating in a cold plasma. Both of these plasma instabilities are capable of generating high intensity magnetic fields and trapping electrons/particles that would emit X-ray radiation spectra similar to ”jitter” radiation. The study of CFI could provide insight to the mechanism for the afterglow of GRBs and the associated radiation spectrum. 1.2.3 Fast Igniter - Inertial Connement Fusion Fusion of hydrogen ion isotopes, proton, deuteron and triton, is one of the mechanisms that powers all stars. A potential future energy source is 8 to create and harness fusion energy, one such concept is inertial confinement fusion, ICF. ICF is a process where nuclear fusion reactions are initiated by compressing and heating a fuel pellet consisting of a mixture of two isotopes of hydrogen, deuterium and tritium, commonly referred to as D- T. The Hydrogen isotope ions fuse together forming a helium nucleus and release an energetic neutron. This small amount of lost mass is converted to a large amount of energy, E = mc 2 . In the conventional ICF concept energy is delivered, typically by multiple high intensity drive lasers, to the outer layer of a spherical D-T fuel pellet that compresses and heats resulting in it exploding outwards. The resulting force causes the interior of the pellet to be driven inward and compress the inner material. In the process shockwaves are created and propagate symmetrically inwards meeting at the center, forming a hot spot of significantly higher density and fusing the ions together. This is called ignition. The energy released by ignition starts a self sustaining thermonuclear burn in the surrounding material, this is called burn. The conventional ICF process is illustrated in Figure 1.2. The conventional ICF concept places very stringent requirements on the symmetry of the fuel pellet, shockwaves must arrive at the same lo- cation, and the distribution of energy contained within the driving lasers, arrive with the same energy at the same time at all points on the pellet with 9 Figure 1.2: Inertial confinement fusion sequence [2]. quasi-perfect spherical symmetry. Lack of symmetry in either component will result in different parts of the fuel pellet reaching maximum density at different times and the pellet will break apart without fusion occurring. A concept which will reduce the stringent requirements of conven- tional ICF is the Fast-Ignitor ICF [51]. In this concept the compression and ignition stages are separated allowing for the fuel pellet symmetry require- ments to be relaxed and the energy of the compression lasers to be reduced 10 by up to an order of magnitude. One such concept is the cone-in-shell where a gold cone is manufactured into the D-T fuel pellet, Figure 1.3 [32]. Figure 1.3: Fast-Ignition ICF with Cone-in-shell concept with gold cone imbedded into a fuel pellet. Inset is X-ray image showing imploded core at tip of cone [32]. The outer layer undergoes the same process as in the conventional ICF concept and a dense plasma forms at the tip of the gold cone. At this point an ultra-short (pico seconds in length) intense laser is focused into the gold cone generating a relativistic beam of hot electrons at the critical plasma surface. These hot electrons propagate to the core, deposit their energy and ignite the fusion process. While the Fast-Ignitor concept reduces the symmetry and drive laser requirements it introduces the requirement for stability of the propagating hot electrons to deposit their energy at the core. Development of an insta- 11 bility could preclude ignition [49]. The CFI in the context of Fast-Ignitor ICF has been the subject of simulation studies [49] and observed experimentally [53]. Thus, understanding CFI is important to the fast-ignitor ICF concept for efficiently propagating the stable beam of hot electrons towards the core of the fusion pellet. 1.3 Thesis Goals and Structure 1.3.1 Goals of Thesis The current filamentation instability has relevance to a broad range of current scientific efforts and to date little experimental research has been conducted to verify and extend knowledge of the instability. This thesis presents and defends the results of an experimental study of the CFI. These results are the first to unambiguously show the occurrence of CFI and study and confirm its main characteristics including: breakup of the beam into multiple filaments, scaling of the transverse filament size with the plasma skin depth, transition from CFI to plasma focusing and suppression of the instability by reducing the growth rate through the beam density. In sup- port of this effort I provide an overview of theory of the current filamenta- tion instability. I study the instability with particle-in-cell simulations where I verify some predictions from theory, explore topics not discussed in liter- ature or theory and provide insight for the experiment and its results. Fi- nally, I present and discuss results from the experiment conducted at the 12 BNL-ATF with a 60MeV electron beam and capillary discharge plasma. In conclusion I provide an idea of what future research on CFI may look like. 1.3.2 Structure of Thesis The remainder of the thesis is structured as follows: Chapter two reviews the current filamentation instability and summarizes previous the- oretical work. Chapter three presents a simulation study of the instability with electron beam and plasma parameters achievable at the BNL-ATF and used in this experiment. Chapter four provides an overview of the ATF experimental facility, experimental setup, specifically the imaging system, and concludes with presenting the measured performance of the imaging system. Chapter five presents and discusses the experimental results which show CFI and confirms theoretical predictions. Chapter six summarizes the thesis and discusses future work that will expand on the study of CFI. Note that the main results, i.e. observation of multiple filaments, scaling of the transverse filament size with the plasma skin depth, transition from plasma focusing to CFI and suppression of CFI by reducing growth rate, were ac- cepted for publication in Physical Review Letters. A further, longer and more detailed publication including material in this text is underway and will be published. 13 1.4 Conclusions Presented motivation for the study of CFI: – In PWFAs CFI can place an upper limit on the accelerating gradi- ent, for a given focused beam size. – The CFI could be the mechanism for generation of large magnetic fields and the emission of radiation observed in the afterglow of GRBs. – Stability of the beam of relativistic hot electrons necessary for the Fast-Ignitor ICF concept and when subject to CFI could preclude ignition. Detailed the goals of this thesis Outlined the structure of the thesis 14 Chapter 2 Theoretical/Technical Review of CFI 2.1 Introduction The ability of a plasma to react collectively allows it to neutralize the charge and current of a beam propagating through it. However, cur- rent neutralization is less efficient than charge neutralization and as a result beam-plasma interactions are susceptible to streaming instabilities, such as the two-stream and current filamentation instabilities. This chapter will de- tail the regime that beam-plasma interactions are susceptible to CFI and the development of and characteristics of the instability. 2.2 Plasma Focusing and Return Currents For this analysis in this section the following assumptions are made 1) the particle position and velocity profiles for the beam and plasma are uniform and 2) particles have only a velocity component in the direction of propagation. A beam propagating in vacuum with no external forces is subject to its self fields. A radial electric field,E br , due to the beams space charge and an azimuthal magnetic field,B b , as a result of the beams current. E br and 15 B b can be obtained from Gauss’s and Ampere’s law respectively and the total force on the beam, the Lorentz force, is: F b =e(E br +v b B b )/ 1 2 b (eE br ) (2.1) wherev b is the velocity of the beam,e is the electron charge and b = (1 v 2 b =c 2 ) 1 2 is the Lorentz, or relativistic factor, and c is the speed of light in vacuum. Equation (2.1) shows a net defocusing force on a beam propa- gating in vacuum. If the beam is relativistic (v b c and b is large) the defocusing forceF b becomes very small. A beam propagating in a neutral plasma is subject to self fields, E br and B b , and external electric, E pr , and magnetic, B p , fields from the plasma. The plasma ions are assumed to be stationary with a mass ratio m i =m e 1; 836, m i and m e are the ion and electron masses. Through the changing magnetic flux of the propagating beam a current of electrons of the plasma is induced with equilibrium driftv b n b =v pe n pe ,n b andn pe are the beam and plasma density respectively andv pe is the velocity of the plasma electrons. By Lenz’s law this current must flow in the direction opposite to the beam current, is called the plasma return current and generatesB p . This return current flows as a mirror of the beam current and is effective at canceling the beam current at scales larger thanc=! pe at all points and not as well for scales smaller than c=! pe . This characteristic response length, c=! pe , is the plasma skin depth [12] and! pe = (n pe e 2 =" o m e ) 1 2 is the plasma 16 electron angular frequency with " o the permittivity of free space. The Lorentz force on the beam due to the fields from the beam and the plasma is: F b =e ((E br E pr ) +v b (B b B p )) (2.2) The force on the beam is thus dependant on the sign and magnitude of the net electric and magnetic fields. Two distinct beam-plasma interaction regimes emerge. A parameter k p r is defined and identifies these regimes, k p = 1=(c=! pe ) is the plasma wave number and r is the transverse size of the beam. k p r reflects the relative transverse size of the beam with respect to the plasma characteristic response length,c=! pe . These two interaction regimes are: 1) Plasma focusing regime – transverse beam size is smaller than the plasma skin depth, r <c=! pe ork p r < 1 2) Return current regime – transverse beam size is larger than the plasma skin depth, r >c=! pe ork p r > 1 In the plasma focusing regime the majority of the plasma return cur- rent flows outside the beam and the current densityJ b >J pe , the beam and plasma electron current densities, which by Ampere’s law leads to B b > B p and a net magnetic field exists,jB b B p j > 0. The net magnetic field focuses the beam, r reduces, and starts a positive feedback loop where as 17 the beam is focused further the net magnetic field increases further. Chen et al. [14] provides an analysis of plasma focusing in the linear regime, n b =n pe < 1, for PWFA’s (see chapter 1). The focusing force, transverse wake- field in [14], depends on the distribution of beam particles, length of the beam and plasma density, but all else equal the focusing force scales with the beam density,F b /n b . In the return current regime the majority of the return current passes through the beam. Ideally the net magnetic field is zero,jB b B p j = 0, and the beam is subject only to the net electric field, this is called current neutralization. Even if not fully neutralized the focusing force is signifi- cantly reduced, relative to the plasma focusing case. In reality there are several factors not included in this simplistic model. Two important factors are 1) the beam particles have a transverse velocity component and 2) the beam and plasma particle position and velocity profiles are non-uniform. The beam particles thus far have been assumed to have only a veloc- ity component in the propagation direction of the beam, however in reality particles also have transverse velocity components. In accelerator physics a measure of the average spread of particle coordinates in position-and- momentum phase space is called beam emittance. The RMS emittance is defined as: x = p <x 2 ><x 0 2 ><xx 0 > 2 (2.3) 18 x is the particle position, x 0 = P x =P z is the angle of the momentum. p <x 2 >, p <x 02 > and < xx 0 > are the rms beam size, angular spread and correlation between the x and x 0 . As the spread in momentum, or velocity, increases the emittance increases. Emittance is often associated to a temperature (see later): 1 2 m e v 2 ? = 2 1 2 k B T e? (2.4) v 2 ? =< x 02 > where k B is Boltzman’s constant and T e? is the temperature of the electrons. The beam emittance is an initial condition for the particles velocity and as such it makes the beam defocus (after a waist) and a focusing force first has to overcome this outward motion. A plasma can collectively respond on the order of the cold plasma skin depth c=! pe . For feature sizes of the beam smaller than c=! pe the plasma will not be able to completely cancel or react i.e. there can be “non canceling” plasma return currents at scales. c=! pe . The second assump- tion made was that the beam and plasma position and velocity profiles are uniform. It is when the beam and plasma profiles are non-uniform that a beam-plasma interaction in the return current regime is susceptible to in- stabilities such as the CFI. Since in reality beams and plasmas are always non-uniform in the transverse density current profiles and the plasma “am- plifies” those at thec=! pe scale. 19 2.3 Plasma Instabilities: Weibel, Two-Stream and Current Filamentation Two specific plasma instabilities, the Weibel and two-stream, are rel- evant to understanding the current filamentation instability and defining the regime favorable to it. The wave vector and electric field orientations for these three plasma instabilities is shown in Figure 2.1. Growth Rate Plot for Beam Density X Z (Beam Propagation - V b ) Y k E Current Filamentation k E Two-Stream k E Weibel (Low Temperature Axis) Figure 2.1: Electric field and wave vector for current filamentation, two- stream and Weibel instabilities. Z-axis is the propagation axis for the beam for the current filamentation and two-stream instabilities and the low tem- perature axis for the Weibel instability. One important note about the theoretical work cited here is that it is for an infinite sized neutral plasma for the three instabilities and infinite sized beam of electrons for CFI and two-stream instabilities. In some cases the authors note that their work should apply to finite sized beams, but do 20 not provide results confirming this or do not point at differences that may be expected. The experimental results presented here after are of course obtained with beams and plasmas of finite extent. 2.3.1 Weibel Instability The Weibel instability [54] is a kinetic instability and is due to anisotropic distribution of electron velocities (or temperatures) in a homo- geneous collisionless plasma. The instability supports unstable growth in transverse electromagnetic waves leading to generation of transverse current filaments. The wave vector is normal to the anisotropic temperature axis, Fig. 2.1. For example, if T ex > T ey = T ez unstable waves develop in they andz planes,T ex;y;z are thermal velocities in thex,y andz directions respectively. It is important to note that the instability will develop without a beam, as in the original derivation by Weibel, but a beam can introduce or amplify the anisotropies. In literature the Weibel instability is often confused with CFI, however CFI is a strictly beam driven instability – sometimes called beam driven Weibel. 2.3.2 Two-Stream Instability A beam propagating in a plasma, with non-uniform transverse cur- rent density profiles, in the return current regime, the return current is the second stream, is susceptible to the two-stream instability [9]. The two- 21 stream instability is an electrostatic streaming instability subject to unstable growth of purely longitudinal waves and is seeded by non-uniformities in the beam and plasma return currents. Its wave vector is parallel to its elec- tric field, Fig. 2.1, and as a result does not generate magnetic fields. The growth rate is [10]: TS = p 3 2 4 3 1 3 b ! pe (2.5) o = v b =c is the relative beam velocity and = n b =n pe is the ratio of beam density to plasma density. 2.3.3 Current Filamentation Instability The seeding mechanism for CFI is the same as the two-stream in- stability [18, 38]. However, the CFI is an electromagnetic instability with a purely imaginary frequency and is subject to unstable growth of purely transverse waves. Hence, the wave vector of the CFI is perpendicular to the electric field, Fig. 2.1, and the beam breaks apart into high current density filaments and enhances magnetic fields. The evolution of CFI can be represented as a three step process. 1) linear growth – the beam breaks apart and filaments develop, 2) saturation – filaments become of the size and spacing of the plasma skin depth, c=! pe and 3) filament coalescence – over time the filaments coalesce or merge together. Figure 2.2 shows the growth in magnetic energy due to CFI from 22 a simulation in Chapter 3 and beam density profiles for the phases of the development of the instability. 10 -11 10 -10 10 -9 10 -8 10 -7 0 1 2 3 4 5 6 7 Magnetic Energy - J Distance - cm A B C L p =0.2cm L p =2.2cm L p =3.5cm Y X 391μ μ μ μm 391μ μ μ μm (b) (c) (d) Beam Density (arb. units) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 L p =4.9cm (e) (a) Figure 2.2: Magnetic energy and beam density evolution, simulations from chapter 3 for “nominal” case, section 3.3.1. Three steps in CFI evolution are denoted A) linear growth, B) saturation and C) filament coalescence. Color map is the same for all beam density images. CFI Development/Process/Evolution Linear growth In the return current regime uniform beam and plasma profiles were assumed and as a result the beam is both charge and current neutralized. The beam generates a return current in the plasma 23 however, non-uniformities that exist in real or simulation beam and plasma profiles lead to unequal and opposite beam and plasma return current densities and to the development of local magnetic fields. These opposite current densities repel each other, locally enhancing magnetic fields and exerting a force on the opposite current. This repulsion increases the non-uniformities in the distribution of the current densities and a positive feedback loop is established leading to breakup of the beam into transverse high current density filaments. The local magnetic fields focus the resulting filaments. Since the CFI generates magnetic fields the growth in magnetic field energy in the beam/plasma volume is chosen to characterize its growth. This is the exponential portion of the magnetic energy curve and represents linear growth of the instability, labeled region A in Fig. 2.2a. Figure 2.2b to c shows the beam density transition from a gaussian beam to breakup of the beam into weak intensity filaments. Saturation The filaments generated in the linear stage are focused and when they are of size and spacing c=! pe the instability saturates. At this point the magnetic energy is no longer growing exponentially and is labeled region B on Fig. 2.2a. The beam density shows filamentation, Fig. 2.2c with one high intensity on axis filament surrounded by filaments of weaker intensity. 24 Coalescence The beam filaments are subject to mutually attractive Lorentz forces. Therefore, filaments coalesce, or merge, into a decreasing number of filaments, until only one filament remains, assuming a long enough plasma. The time between successive coalescing is increasingly farther apart in time and if the filaments were fully current neutralized the time required to coalesce would become exponentially long [33]. These coalesced, or merged, filaments are initially larger than the skin depth but eventually pinch down to c=! pe . Peaks in the magnetic field occur just before coalescing, see Figures 4 and 5 of [39]. Estimates for coalescing time are provided in [4, 33]. The coalescing of the filaments process is seen where the magnetic energy is no longer increasing, labeled region C on Fig. 2.2a, and the filaments are higher intensity and moving towards the high intensity on axis filament from Fig. 2.2d to e. In summary, the development of CFI is characterized by breakup of the beam into high current density filaments and generation or enhance- ment of magnetic fields. 25 Characteristics of CFI Magnetic Fields and Radiation With the large magnetic fields generated by the filaments and with particles effectively trapped inside, one would expect these particles to emit radiation, possibly similar to synchrotron radiation (electrons gyrating in high intensity magnetic fields) or “jitter” radiation in an astrophysical con- text and was discussed in Chapter 1. Radiation was observed in [27]. Growth Rate and Dependence on Plasma Density The growth rate of CFI for a transversely cold beam propagating in a cold plasma is [10]: CFI = o r b ! pe (2.6) Let us delve deeper into the meaning of CFI by expanding Eqn. (2.6): CFI = o s 1 b n b e 2 m e " o = o r 1 b ! pb / p n b (2.7) Equation (2.7) shows that the growth rate scales with the beam density and is independent of the plasma density. While the plasma density does not play a role in the growth of the instability it plays two important roles. First, the plasma density must be such that r c=! pe for the interaction to be in the return current regime otherwise plasma focusing occurs. Second, when 26 CFI is saturated it sets the size and spacing of the filaments which are on the order of the plasma skin depth,c=! pe . Dominant Instability Modes The two-stream and current filamentation instabilities both occur in the return current regime and are seeded by non-uniformities in the beam and plasma profiles. However, the two-stream is a longitudinal instability while the CFI is a transverse instability. So, which instability is dominant? By comparing the growth rates for CFI, Eqn. (2.5), and the two-stream, Eqn. (2.6): CFI TS = o 1 6 p 3=2 4 3 p b (2.8) For a given o and a relativistic beam, b > 1, CFI will be the dominate in- stability while for a non-relativistic beam, b 1, the two-stream instability will dominate provided the bunches are long compared to the characteristic response length of the plasma,k p z 1, z is the rms bunch length. The conclusion that the CFI is the dominant instability for relativis- tic beams assumes there is only two possible instabilities. Bret et al. [10] showed that for a relativistic beam in a cold or warm plasma the dominant or fastest growing mode is a mixture between the two-stream and filamen- tation instabilities, called mixed TS-CFI. This mixed mode has a wave num- ber angle, is the angle of the wave vector from the beam propagation axis, 27 between the extremes of = 0 (two-stream) or = =2 (current filamenta- tion). The growth rate for this mixed mode is [10]: M = p 3 2 4 3 b 1 3 ! pe / 1 1 6 b (2.9) Bret et al. had a second reason for proposing this mixed mode, a pure CFI mode will not result in filamentation. This can be seen from letting 1 and E 1 be the first order perturbations to the electron charge density and electric field. Poisson’s equation in Fourier space is: kE 1 (k;!) = 4 1 (k;!) (2.10) For CFIkE 1 (k;!) = 0 and thus there is no density perturbation in the linear regime. They further showed the mixed TS-CFI instability could generate the necessary linear density perturbations required for filamention. It is interesting to note that M / n 1 3 b n 1 6 pe is dependent on both the beam and plasma density, unlike for CFI which is dependant only on the beam density. In general there is of course always a two-stream and a CFI contribution and the instability wave number is never exactly perpendicular tov b . However, for the relativistic beam case of the experiment described herek?v b , CFI characteristics dominate and no evidence of the two-stream instability was detected. 28 Transverse Temperature Momota et al. [38] first showed that the transverse beam tempera- ture (random particle velocity) can suppress the instability if large enough. Equation (2.6) for the CFI growth rate assumes no transverse temperature, but Bret et al. provided a correction to the CFI growth rate for a transversely warm beam [10]: CFI = o r b 1 T 2 e? b ! pe (2.11) T e? = v ? v b (2.12) T e? is the transverse thermal temperature of the beam andv ? is the trans- verse velocity spread, see Eqn. (2.4). Equation (2.11) shows that as the trans- verse beam temperature increases the growth rate slows and the vanishes when T e? p b . Intuitively this makes sense since the CFI is a focus- ing force and emittance, transverse beam temperature, acts as a defocusing force and the two compete. 2.4 Summary and Conclusion The conditions that are favorable for the development of CFI are: 1) Return current regime where the transverse beam size is larger than the plasma skin depth, r c=! pe ork p r 1 – otherwise plasma focusing regime 29 2) Relativistic beam, b > 1 – otherwise two-stream instability regime 3) Low transverse beam temperature," N – growth rate will be reduced or if large enough will suppress CFI 4) Large beam density, n b – the lower the beam density the slower the growth of CFI Key characteristics of CFI are: Beam will breakup into high current density filaments Filament size and spacing will be of size of the plasma skin depth l c=! pe Generation or enhancement of magnetic fields Particles trapped in large magnetic fields will emit radiation In Chapter 5 results of the experiment confirm, for the first time, some key characteristics of CFI discussed in this chapter. These include filamentation of the beam, transverse filament size scales with the plasma skin depth, transition from plasma focusing to CFI is aroundk p r 1 and suppression of the instability by reducing the beam density. 30 Chapter 3 Simulation Study of CFI The development of the CFI involves the complex interaction between two large populations of particles, those of the bunch and those of the plasma. Simulation tools are a useful and efficient way to gain insight and study this interaction. These tools allow one to probe things which cannot be probed in an experiment (either due to the physical impossibility or because diagnostics are not available) and allow us to study the variation of parameters on an individual basis. The purpose of this chapter is three-fold: 1) verify some key char- acteristics predicted from theory, 2) explore relationships not discussed in literature and 3) provide insight for the experiment as beam and plasma pa- rameters are varied. From theory the predictions for scaling of the growth rate with beam density, enhancement of net magnetic fields and scaling of the transverse filament size with the plasma skin depth will be explored. Not explicitly discussed or predicted from theory are the questions: Does the growth rate scaling hold for a finite sized beam, does the plasma den- sity play a role in the growth rate and what impact does the beam finite emittance have on the instability. Through the chapter insight is provided 31 for what can be expected in the experiment and what variation of the beam and plasma parameters will result in. Finally, one method of seeding the instability is proposed and evaluated. 3.1 Particle-in-Cell Codes for Plasma Simulations For studying the instability the simulation tool QuickPIC [23], de- veloped to study PWFA and LWFA and is applicable to the study of CFI, is used. QuickPIC is a highly efficient, fully parallelized, fully relativistic, three-dimensional particle-in-cell, PIC, code and is based on the quasi-static approximation. The quasi-static approximation assumes that the beam does not evolve in the time it takes the highly relativistic beam to pass a plasma particle, or a thin plasma slice, and thus the evolution of the beam and the plasma occur on different time scales and these time scales can be sepa- rated. This approximation reduces the fully three-dimensional electromag- netic field solver and particle pusher to a two-dimensional (transverse) field solver and particle pusher, where the beam evolves over very large time steps and significantly reduces the computation time. A fully explicit PIC code would resolve the plasma skin depth,c=! pe , where as with the quasi- static approximation only the betatron wavelength of the bunch evolution is resolved, 2 p 2 b (c=! pe ), b is the Lorentz factor. Therefore a savings of the order 2 p 2 b is expected, which can be large for relativistic bunches ( b > 100). 32 Birdsall and Langdon [6] provides an excellent overview of PIC for plasma simulations. The simulation domain is a moving window and the window moves along the simulation length. A fully electromagnetic PIC algorithm first breaks the simulation domain into a mesh. This mesh is uniform in QuickPIC and the charge density, , and current density, J, are defined on the mesh points. Plasma particles are represented by macro-particles that are distributed over the simulation domain and then loaded onto the mesh with their associated current and charge assigned to the nearby mesh points. Faraday’s (3.1) and Ampere’s law (3.2) are discretized with the electric,E, and magnetic,B, fields defined on the mesh points.c is the speed of light in vacuum. 1 c @B @t =rE (3.1) 1 c @E @t = 4 c JrE (3.2) Faraday and Ampere’s equations are time-centered (B andJ, current den- sity, leadE by half a time step, t 2 , as are the particle velocities and positions) [6] for integration purposes and allow for consecutive updates ofE andB respectively. The updated fields are used to advance the particles to new positions,x, and update their velocities,v, via the relativistic equations of motion: @P @t =e E + rB c +F external (3.3) 33 @x @t =e E + P m b (3.4) P is the particle momentum andF external is any external force that may be applied to the beam or plasma particles,m ande are the mass and charge of the particle. Figure 3.1 shows the particle-in-cell computation cycle. PIC Simulation Cycle Update Beam Particle Positions: X i Allocate Charge and Current Density on Mesh: ρ ρ ρ ρ m,n , J m,n Calculate Electric and Magnetic Fields on Mesh: E m,n , B m,n Update Beam Particle Velocities: v i At each time step, for particle i and mesh point (m,n) Figure 3.1: Particle-in-cell computational cycle. Since CFI is an instability that is seeded by noise (non-uniformities) in the beam and plasma profiles, initial noise in the simulation is necessary. In the absence of noise the plasma would generate a return current exactly equal to and cancel the beam current which would result in no CFI. Ran- domness, noise, is provided by the discrete particle distributions in position and velocity that represent the finite bunch temperature (velocity space) and bunch shape (space) and emittance (space and velocity) [6]. 34 3.2 Simulation Parameters The CFI experiment at the ATF used the uncompressed beam with a bunch length z 5 ps. The computational resources necessary to simu- late a bunch of this length (for plasma densities n e > 10 15 cm 3 ) with suf- ficient resolution are beyond the resources available for this work and if they were available would have significantly increased required simulation time (the simulation system would have been significantly greater in size, both spatially and in numbers of particles). Therefore, to study the instabil- ity with simulations parameters that are similar to those achievable at the ATF are used but with the over-compressed beam [30], with a bunch length z 100 fs. CFI is a purely transverse instability and the bunch length is not expected to play any role in its development therefore, simulations with a shorter bunch length are appropriate. Parameters Value Charge –Q (pC) 100-200 Typical Bunch Transverse Waist Size – 0x;y (m) 50 to 100 Bunch Length – z (fs) 100 Bunch Density –n b (cm 3 ) 3 10 14 Energy (MeV) 58 ( 0 = 117) Table 3.1: ATF over-compressed electron beam parameters. The beam density for a tri-Gaussian beam is calculated from: n b = N b (2) 3 2 2 r z (3.5) 35 N b = Q e (3.6) r – bunch transverse waist size, z – longitudinal bunch length,N b – num- ber of bunch particles andQ – bunch charge. Recall the theoretical growth rate of the instability, for an infinite transverse beam size, from Chapter 2 is Eqn. (2.6) and is rewritten here for convenience: = o r o ! p = o s e 2 o m e o p n b (3.7) o – relative velocity of the beam, m e – electron mass, o – permittivity of free space. With the beam parameters in Table 3.1, with Q = 112 pC and r = 75m, and Eqn. (3.7) the growth length is estimated to be: c = 3:53 mm A plasma of lengthL p = 2 cm, the available capillary length at the ATF, is estimated to provide greater than 5 e-foldings. The ATF beam is relativistic and if an appropriate plasma density is selected such that r c=! p (in the case for r = 75m andn e > 1 10 17 cm 3 , r > 4:5c=! p ) the beam-plasma interaction is in a regime favorable to study CFI. The “nominal” simulation parameters are shown in Table 3.2. 3.3 Simulations Results The study will start with the “nominal” case and then individually change some beam and plasma parameters to study their effect on the insta- 36 Parameters Value Value Simulation Box –X (m; # of grids) 1,100 512 Simulation Box –Y (m; # of grids) 1,100 512 Simulation Box –Z (m; # of grids) 160 128 Plasma Particles/Cell 1 3-D Time Step (m) 14.7 Beam Particles –X (#) 512 Beam Particles –Y (#) 512 Beam Particles –Z (#) 128 Lorentz Factor – 0 117 RMS Beam Transverse Waist Size – r (m), r =(c=! p ) 75 10 RMS Bunch Length – z (m) 30 Charge –Q (pC) 112 Normalized Emittance –" N (mm mrad) 5 Plasma Density –n e (cm 3 ) 5 10 17 Skin Depth –c=! p (m) 7.5 Capillary/Plasma Length –L p (cm) 2 Table 3.2: “Nominal” simulation beam and plasma parameters. bility. The study will include evolution of the beam density and magnetic fields (net fields of beam and plasma), size of filaments and growth rate of the instability. 3.3.1 Simulations with \Nominal" Parameters To provide a starting point for the instability development the evolution of the beam for the ”nominal” parameters from Table 3.2 is evaluated. In the CFI regime the initially tri-gaussian beam will break apart into narrow high current density filaments of size and spacing on the order of the plasma skin depth. The development of these high 37 current density filaments will enhance the net magnetic fields and hence the magnetic energy. Figure 3.2 shows the transverse integrated (summed alongz-direction) beam density for nine propagation lengths in a uniform plasma of densityn e = 5 10 17 cm 3 (plasma skin depth isc=! p = 7:5m). The color map for all nine images is the same. At L p = 0:2 cm, Fig. 3.2a, breakup of the beam or filamentation has not occured and can be considered the same asL p = 0:0 cm, the plasma entrance. Filamentation is observed atL p = 2:2 cm, Fig. 3.2c, and atL p = 3:5 cm, Fig. 3.2f, the average root mean squared, rms, filament size is 8m. From L p = 3:5 cm, Fig. 3.2g, toL p = 6:8 cm, Fig. 3.2i, the filaments are attracted to each other and move closer together before finally merging/coalescing. At L p = 6:8 cm fewer filaments are present with one filament of significantly higher beam density, all filament sizes are 8m. To gain a feeling of how dramatic the increase in the beam density is from the incoming beam through to filamentation and saturation of the instability Figure 3.3 shows 3-dimensional plots of the beam density at four plasma lengths, L p = 0:2; 2:2; 3:1 and 4:0 cm. The peak beam, or filament, density (z-axis) increases by almost a factor of 20, from 0:03 to 0:62 arb:units The evolution of the instability is also observed through the growth in magnetic energy in the system. The magnetic energy is given by: W B = 1 2 0 Z V B 2 dV (3.8) 38 L p =0.2cm L p =1.8cm L p =2.2cm Y X 391μ μ μ μm 391μ μ μ μm L p =2.7cm L p =3.1cm L p =3.5cm L p =4.0cm L p =4.9cm L p =6.8cm (a) (b) (c) (d) (e) (f) (g) (h) (i) Beam Density (arb. units) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Figure 3.2: Integrated beam density showing the evolution of CFI for “nom- inal” beam in a uniform plasma density n e = 5 10 17 cm 3 . Note that L p = 0:2 cm can be considered the same asL p = 0:0 cm as CFI has not yet developed. Color map is the same for all images. 0 is permeability of free space andV is the volume of simulation box. Eqn. (3.8) can be discretized to: W B = 1 2 0 X MeshPointsx;y;z (B 2 x +B 2 y )V cell (3.9) V cell is the volume of a simulation cell. The magnetic energy in the simu- lation box is calculated at each time step and is shown in Figure 3.4. The 39 0.5 0.4 0.3 0.2 0.1 0 215 322 391 0 391 107 215 322 107 Y (μ μ μ μm) X (μ μ μ μm) Beam Density (arb. units) L p =0.2cm L p =2.2cm (a) (c) 0.5 0.4 0.3 0.2 0.1 0 215 322 391 0 391 107 215 322 107 Y (μ μ μ μm) X (μ μ μ μm) Beam Density (arb. units) 0.5 0.4 0.3 0.2 0.1 0 215 322 391 0 391 107 215 322 107 Y (μ μ μ μm) X (μ μ μ μm) Beam Density (arb. units) L p =3.1cm L p =4.0cm (e) (g) 0.5 0.4 0.3 0.2 0.1 0 215 322 391 0 391 107 215 322 107 Y (μ μ μ μm) X (μ μ μ μm) Beam Density (arb. units) Figure 3.3: Integrated beam density of the evolution of the beam/instability for “nominal” beam parameters in plasma density ofn e = 510 17 cm 3 . Let- ters denote image corresponding to those in Fig. 3.2. Note thatL p = 0:2 cm can be considered the same asL p = 0:0 cm as CFI has not yet developed. enhancement in magnetic energy, due to the development of CFI increases by two orders of magnitude from the entrance to the plasma to saturation of the instability. Each vertical dashed line and corresponding letter refers to an image in Fig. 3.2. The magnetic energy starts to exponentially increase at L p 2:2 cm and in Fig. 3.2c a weak filamentary structure in the beam density is observed. The exponential growth in magnetic energy concludes aroundL p 3:5 cm and corresponds to saturation of CFI. ForL p > 5 cm the 40 curve is essentially flat and corresponds to coalescence of filaments seen in Fig. 3.2. Growth Rate Plot for Beam Density 10 -11 10 -10 10 -9 10 -8 10 -7 0 1 2 3 4 5 6 7 Magnetic Energy - J Distance - cm a b c d e g h i f Figure 3.4: Magnetic energy in simulation box versus plasma length for “nominal” case. Vertical lines and letters correspond to those of Fig. 3.2. Figure 3.5 shows the transverse integratedX-component of the net magnetic field,B x +B y , for the same propagation distances as in Fig. 3.2. The resulting high current density filaments enhance the net magnetic field by 70 times, from 0:003 to 0:208 arb:units. The largest magnetic fields are around the highest density (and hence current density) filaments, as is ex- pected. 3.3.2 Function of Beam Density From theory, for an infinite transverse beam size, the growth rate of the instability, Eqn. (3.7), increases (decreases) as the beam density is 41 L p =0.2cm L p =1.8cm L p =2.2cm Y X 391μ μ μ μm 391μ μ μ μm L p =2.7cm L p =3.1cm L p =3.5cm L p =4.0cm L p =4.9cm L p =6.8cm (a) (b) (c) (d) (e) (f) (g) (h) (i) Magnetic Field (arb. units) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Figure 3.5: Integrated x and y magnetic fields (B x +B y ) for the “nominal” beam parameters in a uniform plasma density n e = 5 10 17 cm 3 . Color map is the same for all images. increased (decreased), while holding all other beam and plasma param- eters constant. In simulations the beam density is changed through the beam charge, holding all other parameters constant. Figure 3.6 shows a plot of the magnetic energy on a logarithmic scale as a function of propagation distance in a uniform density plasma for three beam charges, Q = 56; 112 and 225 pC. The growth rate of the instability is the slope 42 from the exponential fit to the linear portion of each curve, the exponential fits are shown as solid lines. Visually, the growth rate increases with the beam density. Additionally, saturation is reached (vertical dashed lines) at shorter distances as the beam density is increased. Note that for the lowest density case (Q = 56 pC) saturation is not reached over 4 cm of plasma, one could say that CFI can be suppressed over a fixed plasma length by reducing the growth rate, for exampleL p = 2:0 cm . Growth Rate Plot for Beam Density 10 -11 10 -10 10 -9 10 -8 10 -7 0 1 2 3 4 Magnetic Energy - J Distance - cm Q=225pC Q=112pC Q=56pC Figure 3.6: Magnetic energy over 4 cm of uniform plasma densityn e = 5 10 17 cm 3 for three beam chargesQ = 56; 112 and 225 pC. Dashed vertical lines represent approximate saturation length. Saturation is not reached forQ = 56 pC. Solid lines are fits to the exponential growth and the slope represents the growth rate. The beam density atL p = 2:5 cm is shown in Figure 3.7 for all three beam charges in Fig. 3.6. Here the instability forQ = 225 pC has approxi- mately saturated,Q = 112 pC is still developing andQ = 56 pC the instabil- 43 ity has not started to develop, i.e. has been suppressed. Q=56pC Q=112pC Q=225pC X 391μ μ μ μm Q=56pC Q=112pC Q=225pC Y X 391μ μ μ μm 391μ μ μ μm 0 0.05 0.1 Beam Density (arb. units) Figure 3.7: Integrated transverse beam density atL p = 2:5 cm with plasma densityn e = 5 10 17 cm 3 and otherwise “nominal” beam parameters for three beam charges. Color map is the same for all three images. The theoretical growth rate expressed by Eqn. (3.7) is for an infinite transverse size and predicts that the growth rate scales with the beam den- sity, / p n b . Figure 3.8 plots the growth rate from simulations, derived from the growth in magnetic energy, and Eqn. (3.7) for the three beam den- sities, beam charges, in Fig. 3.6. The power fit for the growth rate from the simulation results shows an exponent of 0:46 and confirms that the theoreti- cal growth rate scales with the beam density to the 0:5 power. This shows that the scaling in the infinite case also holds for the finite case. Note that there is a difference of 15% between the theoretical growth rate and the one obtained from simulations. 3.3.3 Function of Plasma Density The plasma density, for a given transverse beam size, controls the beam-plasma interaction type. The parameterk p r (k p = 1=(c=! p ) – plasma 44 Scaling of Filament Size with Skin Depth 1.5 2.0 2.5 3.0 3.5 4.0 4.5 1 10 14 2 10 14 3 10 14 4 10 14 5 10 14 6 10 14 Growth Rate (Γ Γ Γ Γ) - cm -1 Beam Density (n b ) - cm -3 Simulations: Γ Γ Γ Γ=8.07x10 -7 (n b ) 0.46 Figure 3.8: Growth rate from theory (red line) and simulations (blue points) for three beam charges. Solid blue line is the power fit to the data points. wave number) is defined to identify these interactions: CFI (k p r 1) and plasma focusing (k p r . 1). Additionally, when in the CFI regime the plasma density will control the size of the filaments which will in turn control the number and spacing of the filaments. The instability is saturated when the filament size is on the order of the plasma skin depth. Therefore, when in the CFI regime and satu- rated a linear relationship between the plasma skin depth and the trans- verse filament size is expected. Figure 3.9 shows the rms filament sizes for the “nominal” beam parameters at L p = 4 cm and for four plasma den- sities (skin depths) n e (c=! pe ) = 3:54 (8:9); 5:00 (7:5); 7:07 (6:3) and 10:00 10 17 cm 3 (5:3m). A linear fit, solid red line, to the filament sizes shows an approximately linear relationship, a slope of 1:01. Filament sizes are not all 45 the same since different charge (density) is contained in each filament that governs the growth rate. Scaling of Filament Size with Skin Depth 2 4 6 8 10 12 5 6 7 8 9 Filament Size - σ σ σ σ fil (μ μ μ μm) Plasma Skin Depth - c/ω ω ω ω p (μ μ μ μm) Simulations: σ σ σ σ fil =0.59 + 1.01 (c/ω ω ω ω p ) Figure 3.9: Transverse rms filament sizes at L p = 4 cm for plasma skin depths c=! pe (n e ) = 5:3m (1:00 10 18 cm 3 ); 6:3m (7:07; 7:5 (5:00 10 17 cm 3 ); and 8:9m (3:54 10 17 cm 3 ). Red line is fit of data points and black line shows linear scaling. As the size of the filaments decreases with increased plasma density the number of filaments increases and leads to the peak filament density decreasing (more filaments to spread the total beam charge over). Figure 3.10 shows that as the plasma density increases the peak beam density de- creases. Theory, Eqn. (3.7), predicts that the CFI growth rate is independent of the plasma density, for the infinite transverse case. Figure 3.11 shows the logarithm of the magnetic energy for the same four plasma densities over plasma length of L p = 4 cm. Visually, the slopes decrease with in- 46 Scaling of Filament Size with Skin Depth Increasing n e 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 5 6 7 8 9 Peak Beam Density - A.U. Plasma Skin Depth - c/ω ω ω ω p (μ μ μ μm) Figure 3.10: Peak beam density for plasma skin depth (plasma density) c=! pe (n e ) = 5:3 (10:00); 6:3 (7:07); 7:5 (5:00); and 8:9m (3:54 10 17 cm 3 ). creasing plasma density, showing there is a dependence and it can be sig- nificant. Note that the initial, L p = 0:0 cm, magnetic energy is lower for higher plasma densities and represents the return current cancelation abil- ity of the plasma, i.e. more return current is passing through the beam at higher plasma densities. As the plasma density is reduced such thatk p r 1, holding r con- stant, the beam-plasma interaction transitions from CFI to plasma focusing. In the latter regime the beam does not break apart into filaments but leads to a single on axis feature, i.e. the focused beam, and the focused beam size does not scale with the plasma skin depth. Figure 3.12 shows the transverse rms filament size and focused beam size at L p = 4:0 cm for the four high 47 Growth Rate Plot for Plasma Density 10 -11 10 -10 10 -9 10 -8 10 -7 0 1 2 3 4 Magnetic Energy - J Distance - cm n e =7.1x10 17 cm -3 n e =5.0x10 17 cm -3 n e =3.5x10 17 cm -3 n e =1.0x10 18 cm -3 Figure 3.11: Magnetic energy for four plasma densities and “nominal” beam parameters. Dashed vertical lines represent approximate CFI saturation length. plasma densities of Fig. 3.9 (in the CFI regime) and three additional lower plasma densities which place the beam-plasma interaction in the focusing regime, n e (c=! pe ) = 0:5 (75:0); 1:0 (53:0) and 2:0 10 17 cm 3 (37:5 m). These three lower plasma densities have only single features, the focused beam, and their focused transverse size does not scale with the plasma skin depth. Table 3.3 shows that for 1k p r 2 the interaction is in the plasma focusing regime and for k p r > 8 the interaction is in the CFI regime and summarizes the growth rate and average rms transverse feature size for the seven plasma densities studied. 48 Scaling of Filament Size with Skin Depth 0 5 10 15 20 0 10 20 30 40 50 60 70 80 Filament Size - σ σ σ σ fil (μ μ μ μm) Plasma Skin Depth - c/ω ω ω ω p (μ μ μ μm) Focusing CFI Increasing n e Figure 3.12: Filament sizes, rms, for plasma densities (skin depths) in Fig. 3.11 and three lower plasma densities c=! p = 37:5 (2:0 10 16 cm 3 ); 53:0m (1 10 16 cm 3 ) and 75:0m (5:0 10 15 cm 3 ). Solid line shows ideal scaling of CFI. Plasma Plasma Average RMS Density Growth Rate Skin Depth Filament Size n e (cm 3 ) (mm 1 ) Regime k p r c=! p (m) l (m) 5:0 10 15 – Focusing 1.0 75.0 12.8 1:0 10 16 – Focusing 1.4 53.0 12.3 2:0 10 16 – Focusing 2.0 37.5 11.9 3:54 10 17 4.0 CFI 8.4 8.9 9.7 5:0 10 17 3.7 CFI 10.0 7.5 8.0 7:07 10 17 2.0 CFI 11.9 6.3 6.8 1:0 10 18 1.9 CFI 14.2 5.3 5.8 Table 3.3: Summary of growth rate, regime and average rms filament size as the plasma density increases for “nominal” beam parameters. 3.3.4 Function of Beam Emittance As discussed in Section 2.2 the beam emittance (expanding/defocusing) competes with CFI (pinching/focusing) and impacts two characteristics of 49 the instability. First, when the CFI is saturated the transverse filament sizes are c=! p however, the incoming beam emittance imposes a limit on the minimum filament size. Second, the growth rate of CFI is reduced due to the competition between the two forces. Note that the theoretical growth rate, Eqn. (3.7), was derived for a beam with no transverse temperature, or emittance. By increasing (decreasing) the incoming emittance the growth rate of CFI should decrease (increase) and if sufficiently large the beam emittance is expected to neutralize (suppress) the instability. Figure 3.13 shows the magnetic energy over a propagation distanceL p = 4:0 cm for four normal- ized beam emittances," Nx;y = 1; 4; 8 and 12 mmmrad. Visually the slope, growth rate, generally decreases with increased emittance. The saturation length, vertical dashed lines, is shorter for lower emittance, as is expected. In fact, for N = 12 mmmrad the growth rate is sufficiently slowed and does not reach saturation overL p = 4:0 cm. Figure 3.14 shows the integrated beam density atL p = 4 cm of propa- gation for the four emittances in Fig. 3.13. For Fig. 3.14" Nx;y = 8 mmmrad there is one high intensity on-axis filament and four weak off-axis filaments, at 11, 2, 5 and 8 o’clock and for " Nx;y = 12 mmmrad there is no filamen- tation, the instability has been suppressed. As the emittance is increased the filament size increases. For Fig. 3.13 " Nx;y = 1 mmmrad and " Nx;y = 50 Growth Rate Plot for Emittance 10 -10 10 -9 10 -8 0 1 2 3 4 Magnetic Energy - J Distance - cm ε ε ε ε N =8mm-mrad ε ε ε ε N =4mm-mrad ε ε ε ε N =1mm-mrad ε ε ε ε N =12mm-mrad Figure 3.13: Magnetic energy overL p = 4:0 cm of uniform plasma density n e = 510 17 cm 3 for four normalized emittances and otherwise “nominal” beam parameters. Dashed vertical lines represent approximate saturation lengths. Solid lines indicate the growth rate. 4 mmmrad both cases saturate at approximately the same length and at L p = 4:0 cm have approximately the same magnetic energy yet the filament sizes are clearly larger for" Nx;y = 4 mmmrad. QEB Images – 2700 as function of emittance Crop: 1.65 .43 1.41 .69 ε ε ε ε N =1 mm-mrad Y X 430μ μ μ μm ε ε ε ε N =4 mm-mrad ε ε ε ε N =8 mm-mrad ε ε ε ε N =1 mm-mrad Y X ε ε ε ε N =4 mm-mrad ε ε ε ε N =8 mm-mrad 391μ μ μ μm 391μ μ μ μm ε ε ε ε N =12 mm-mrad 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Beam Density (arb. units) Figure 3.14: Integrated transverse beam density atL p = 4 cm for “nominal” beam and plasma parameters for normalized emittances of Fig. 3.13. There are four very weak filaments for " Nx;y = 8 mmmrad and only a single feature for" Nx;y = 12 mmmrad. Color maps are the same for all images. 51 Table 3.4 shows the growth rate and the average rms filament size forn e = 5 10 17 cm 3 and the four normalized emittances in Fig. 3.13 and atL p = 4 cm. With larger emittance the average transverse filament size is larger and in general the growth rate is lower. Note that the magnitude of the emittance allows the filament size to be smaller or larger than the skin depth. Normalized Plasma Average RMS Emittance Growth Rate Skin Depth Filament Size " Nx;y ( mmmrad) (mm 1 ) c=! p (m) l (m) 1 2.7 7.5 4.0 4 3.6 7.5 5.3 8 3.0 7.5 12.2 12 1.7 7.5 12.8 Table 3.4: Growth rate and average transverse filament rms sizeL p = 4 cm for four normalized emittances and otherwise “nominal” beam and plasma parameters. 3.4 Method for Seeding of CFI Through the Initial Beam Prole To this point the non-uniformities that seed the instability (features that have Fourier components atc=! pe ) are due to the initial noise in the par- ticle position and velocity originating from the numerical initialization of the particles distributions. The instability can also be seeded by introducing non-uniformities in the beam or plasma profile. In reality, introduction of seeding into the plasma is difficult, however introducing non-uniformities 52 into the beam profile or charge and current distribution is easier to accom- plish. One method is into introduce sharp features to the beam profile either through increasing the momenta of a subset of the beam particles or com- pletely removing beam particles in a pattern. By imposing sharp features (the transition is small compared toc=! pe ) onto the beam profile the plasma return current profile can not react on this scale and leads to locally unequal and opposite current densities and to seeding of the CFI. To translate this induced momentum distribution to a spatial pattern on the beam profile the particles/beam must be propagated over a distance. This spatial pattern be- comes a density perturbation on the beam profile and provides a structure for the instability to grow from (location of filament formation). The effectiveness of this seeding mechanism is studied with the ATF overcompressed electron beam and plasma. The transverse momenta (P x and P y ) of the particles in a 10m width cross pattern (an “X” - see Fig. 3.15) are increased and then propagate 1 m in vacuum to impose the spatial pattern on the beam density/profile before entering the plasma. Figure 3.15 shows the summed (integrated) transverse bunch density profile after propagation lengths of L p = 0:1; 2:0 and 3:1 cm in a uniform plasma of densityn e = 2:5 10 17 cm 3 for four cases: a) un-modified beam, and for the transverse momentum (P x and P y ) of the particles along the 10m wide cross increased by b) p 10 times, c) 10 times and d) particles 53 are removed. CFI has not developed atL p = 0:1 cm and can be seen as the plasma entranceL p = 0:0 cm. Note that the beam density perturbation from the increased momentum is not visible atL p = 0:1 cm for cases b and c. In the case of the unperturbed bunch of Fig. 3.15a, the filaments pat- tern is random, originating from the noise in the bunch and plasma charge distributions from the simulation particle initialization. When the cross pat- tern is introduced (Figs 3.15b to d) the instability grows preferentially at the locations where sharp features are introduced, namely at the four corners of the cross. The filaments are visible after a shorter plasma length and the CFI is clearly seeded by the sharp spatial features in the bunch profile. In addition, the number of filaments is only four and filaments seem not to ap- pear in the outer regions of the profile (when compared to Fig. 3.15a). Also, the filaments are clearly more intense in the seeded cases, especially in Fig. 3.15d. Although the density perturbation in cases b and c is not visible the perturbation is significant enough to seed the instability. 54 0.1cm 2.0cm 3.1cm Y X 323μ μ μ μm 323μ μ μ μm a b c d Figure 3.15: Transverse summed beam density for propagation distances of L p = 0:1; 2:0 and 3:1 cm in a uniform plasma of densityn e = 2:5 10 17 cm 3 for a beam that is a) un-modified and for increased particle momentum (P x andP y ) of particles in 10m width cross increased by b) p 10x, c) 10x and d) particles removed. Color maps are the same for all images. 55 Figure 3.16 shows the the magnetic energy for the same four cases as in Fig. 3.15. Saturation of the instability is reached at shorter distances (shown as vertical dashed lines) with increased particle momenta, confirm- ing what is observed in Fig. 3.15 and demonstrating the effectiveness of the instability seeding. For the case where particles are removed the initial energy is higher, yet the total charge is lower. This is due to less current cancelation by the plasma. Growth Rate Plot for Emittance 10 -10 10 -9 10 -8 0 0.5 1 1.5 2 2.5 3 Magnetic Energy - J Distance - cm 10x √ √ √ √10x Un-modified Infinite Figure 3.16: Magnetic energy in the simulation box as a function of prop- agation distance in a uniform plasma densityn e = 2:5 10 17 cm 3 for the unmodified beam (red symbols) and particles momenta along 10m wide cross pattern increased by p 10x (blue symbols), 10x (green symbols) and infinity - particles removed - (purple symbols). Slope of fitted solid lines represent growth rate. Dashed vertical lines represent approximate satura- tion distance. Table 3.5 shows that the three seeded cases have higher growth rates, by up to 20%, and shorter growth lengths than the original unseeded case. This shows that introducing sharp features in the incoming bunch trans- verse profile effectively seeds the CFI, even though in the last case charge is 56 removed from the bunch. Emittance Increase Due To Growth Rate Growth Length Wire Mask (cm 1 ) (cm) 0x (un-modified) 1.70 0.59 p 10x 1.79 0.56 10x 1.78 0.56 Infinite (particles removed) 2.15 0.46 Table 3.5: Growth rate and growth length of CFI with 10m wide cross for increased intersected particle momenta (see Fig. 3.16) 3.5 Chapter Conclusions Confirmed the general assumption of the growth rate scaling is similar for the finite and infinite transverse sized cases. Confirmed the growth rate scales with the beam density to the 1 2 power. There appears to be a scaling of the growth rate with the plasma den- sity, not indicated in theory. Transverse filament size scales with plasma skin depth, when in CFI regime, and the emittance sets a minimum size for the filaments. As the plasma density is decreased, holding the transverse beam size constant the beam-plasma interaction transitions from the CFI to the plasma focusing regime. 57 The growth rate of CFI is reduced as emittance of the incoming beam increases. Showed one method of externally seeding CFI through introduction of sharp features imposed on the beam profile. 58 Chapter 4 Experimental Setup 4.1 Experimental Components and Setup In this chapter I will review the primary components of the exper- iment and detail the setup and demonstrate the performance of the direct imaging system. The CFI experiment was performed at Brookhaven Na- tional Laboratory’s Accelerator Test Facility, BNL-ATF. In the experiment we directly image the beam at the capillary exit with minimum feature size and spacing of 10m. Components of the experimental system and setup are shown in Figure 4.1. The ATF 1) 60 MeV linear accelerator with its beam focused at the entrance to a 2) capillary discharge plasma source where 3) optical transition radiation, OTR light, is generated at the capillary exit and the beam is 4) stopped with a tungsten cone. The OTR light is collected with a 5) microscope objective, 6) transported 7) out of vacuum through a glass window, 8) relayed to 9) a camera lens and focused 10) onto a high sensitivity electron multiplying charge-coupled-device, EMCCD, camera. 4.1.1 ATF Facility and 60 MeV Linear Accelerator The ATF linear accelerator consists of a photcathode rf gun [45] fol- lowed by a conventional 2:856 GHz (S-band) accelerator producing a 5 ps- 59 (2) Capillary (5) Microscope Objective (10) EMCCD Camera (3) Si/Au Window H 2 Gas 15kV OTR (6) 1” Turning Mirror (1) 60 MeV Linac e - (4) Tungesten Beam Stop (7) Vacuum Window (8) 3” Turning Mirrors Vacuum Atmosphere (9) 135mm Lens Figure 4.1: Experimental setup: electron bunch from a 60MeV linac is fo- cused to its waist at the entrance to the capillary. The plasma is generated by puffing H 2 gas at 110 Torr into a 1 mm diameter by 2 cm long capillary with an applied voltage of 15 kV. OTR is generated from the bunch/filament electrons and the Au coating. The resulting light is collected and magnified by a microscope objective and relayed with turning mirrors out of the vac- uum chamber then focused onto an EMCCD camera. The camera, lens and closest 3” turning mirror are on a plane lower than the other components, Fig. 4.6. long (rms), 1 nC, 60 MeV single electron bunch with a minimum normalized emittance of 1mmmrad. The bunch is then propagated in vacuum from the linac to beam line 1, Figure 4.2. The beam-plasma interaction location is the center of the Compton chamber where the bunch is focused to its waist 60 size, between 40 and 100m rms. The bunch charge and transverse size are well controlled and variable. Picture/Figure of ATF Layout Ref: http://www.bnl.gov/atf/core_capabilities/images/ATF_Layout.pdf Electron Beam Path Beam Line 2 Beam Line 3 Optical Table/ Camera Compton Chamber/ Capillary Beam Line 1 Figure 4.2: Layout of BNL-ATF experimental hall with the three beam lines. The experiment interaction point is the center of the Compton chamber on beam line one [3]. 4.1.2 Plasma Source and Density Diagnostic The plasma source is a capillary discharge [29] capable of uniform plasma densities from 10 13 to 7 10 17 cm 3 . This source has been used in prior PWFA experiments at the ATF [26, 41, 44]. Hydrogen gas, with a back- ing pressure of 110Torr, is puffed into a 1 mm inner diameter 2 cm long capillary with 15 kV of applied voltage. The capillary assembly is shown in Figure 4.3. 61 Pictures of experimental setup Downstream Electrode / Window Holder Fiber Cables for Plasma Light Electron Beam Path Downstream Electrode / Window Holder Capillary Exit/ Si-Au Window (b) (a) Figure 4.3: (a) Capillary assembly from downstream side, small hole in win- dow holder is capillary exit, 1 mm diameter and (b) capillary assembly in experimental configuration. The plasma discharge light at three locations along the capillary is collected and guided with optical fibers, see Fig. 4.3b, into a spectrometer that measures the hydrogen BalmerH linewidth and the plasma density is derived from Stark broadening [5]. These densities were shown to be within 10% of each other [40]. In the experiment the plasma density is selected by choosing the delay between when the plasma is generated and the electron beam arrives, a longer delay corresponds to lower density. 4.1.3 Direct Imaging System The most direct method to observe CFI is to image the beam at its exit from the plasma (in this case the exit of the capillary) and to observe 62 filamentation of the beam. This is possible with the ATF beam due to the beams low energy density, 60mJ=pulse and is realized by placing a sili- con, Si, window coated with gold, Au, at the capillary exit. The gold coating acts as a radiator generating OTR light. The OTR light is imaged onto an EMCCD camera to obtain the beam/filament transverse profile. 4.1.3.1 OTR Light and Window Transition Radiation Transition radiation [19] is created when charged particles cross a boundary with different dielectric constants,". The particles emit radiation with a particular angular distribution, peaking at 1= , polarization and spectrum, see Figure 4.4. This radiation is generated in both the forward and backward (reflected) directions. Transition Radiation - Figure Medium 1 ε ε ε ε 1 = ε ε ε ε Medium 2 - Vacuum ε ε ε ε 2 = 1 e - Figure 4.4: Transition radiation generated (forward and backward) from an electron propagating from medium 1," 1 =", to medium 2 - vacuum," 2 = 1. 63 The spectral intensity, I 2 , of the forward radiation for a relativistic particle from medium 1, " 1 = " , into medium 2 - vacuum, " 2 = 1 in a frequency ranged! and into a solid angled is [20]: d 2 I 2 d!d = e 2 2 c p " 1 1 p " 1 + 1 2 2 2 + 1 2 2 (4.1) is the angle of radiation emission. Maximization of the spectral intensity occurs when the second term on the RHS of Eqn. (4.1) approaches unity, large" 1 , and is accomplished through material selection. Window/OTR Radiator Design The OTR radiator is placed at the capillary exit and provides a sharp plasma density profile. This is necessary because simulations show the filaments rapidly expand to greater than the filament spacing in the re- duced plasma density expected outside of the capillary and will rejoin into a single filament. After the plasma, the filaments also rapidly expand in vacuum due to their small transverse size and large emittance. In addition, the radiator must not significantly scatter the filaments and withstand the backing gas and plasma pressure. To meet these requirements a silicon sub- strate 100m thick with a 200 nm thick gold coating was selected. 64 Gold was selected for the radiator because it does not oxidize, it is a common process to evaporate onto an Si substrate and its plasma fre- quency is sufficiently high that it should have a good transition radiation photon yield in the visible spectrum. Note that if the window was 100m thick gold it would have significantly scattered the filaments distorting the experimental results. The Si window thickness was determined by balancing three factors: 1. The window must be thick enough to prevent rupture from the gas and plasma discharge. Testing of a 100m window up to pressures greater than 3:5x the typical operating pressure (110 Torr) was con- ducted in a test chamber and the window survived with no signs of damage. 2. Minimize emittance growth of the filaments due to scattering in the window. The resulting emittance growth," g , of the filaments, of diameter l = c=! pe , is determined by: " g = l scat = c ! pe scat (4.2) where the scattering angle, scat , is calculated from [15]: scat = 13:6 MeV cp z r x X o 1 + 0:38 ln x X o (4.3) 65 wherec is the velocity of the beam,p is the momentum of the beam, z is the charge number of the particle (z = 1 for electrons), x is the thickness of the material, X o is the radiation length for the window material or gold layer. Table 4.1 shows the emittance growth due to the Si window and the resulting increase in filament size at the window exit for several win- dow thicknesses using (4.2) and (4.3). The selected 100m thick Si and 200 nm thick gold window will result in 0:7m increase in the filament size and is small compared to the smallest filament sizes ex- pected in the experiment, on the order ofc=! p 10m at the highest plasma density. Table 4.2 also shows that the increase in filament size for a 100m thick Au window is significant and not a viable choice. 3. Minimize window deformations from the gas and plasma discharge to maintain focus of the imaging system on the OTR screen. Table 4.1: Growth in filament size due to scattering from Si window for filament size =10:6m. Radiation lengthX o : Si - 21:82 g=cm 2 [15] Emittance Growth/Increase Thickness (m) Filament Size (m) 50 0.38 100 0.56 150 0.70 200 0.82 66 Table 4.2: Growth in filament size due to scattering from Au layer for fila- ment size =10:6m. Radiation lengthX o : Au - 6:46 g=cm 2 [15] Emittance Growth/Increase Thickness (nm) Filament Size (m) 100 0.08 200 0.11 300 0.14 400 0.17 100,000 (100m) 3.46 When gas flows into the capillary and the plasma discharge occurs, the window could deform due to the pressure difference on the two sides of the window (gas/plasma and vacuum). To image the small filaments onto a CCD, a high magnification system must be employed. A high magnifi- cation system has a small depth of field, where small deformations in the window could blur the filament image. Interferometric measurements in a test environment indicated that for a 100m window the deformations are a fraction of a micron. No issues were identified in the experiment. 4.1.4 Imaging System Design and Components Design of the direct imaging system needed to account for and bal- ance: 1. The capability to resolve feature sizes of 10m 2. Low photon yield from OTR 3. Vacuum compatible optical components 67 4. Alignment of the imaging system For resolution of minimum feature sizes 10m a two lens system and an EMCCD camera with a pixel size 13 13m (ProEM: 1024B, see Ap- pendix A for data sheet) were selected. The two lens system consisted of a 15x microscope objective with a 13:35 mm focal length (Ealing 25-0506-000, see appendix B for specifications) and a 135 mm fixed focal length camera lens (Cannon telephoto EF 135 mm f/2.0L USM Autofocus Lens). The re- maining optical components were optical quality and consisted of 1” and 3” turning mirrors with=20 surface flatness and a glass vacuum window with =10 surface flatness. This combination led to high magnification,M > 10x, sub 10m resolution and was not pixel limited (see next section). The low photon yield from OTR was addressed with a high quan- tum efficiency EMCCD camera, maximizing light collection with an f/2.0 camera lens and removing stray, non-experiment generated, light. Light shielding surrounded all optical components and lights in the experimental hall were turned off. The front mirror of the microscope objective is made of Aluminum and is also an OTR radiator. A tungsten cone, Figure 4.5, with height 15 mm and base the diameter of the front mirror, was mounted to the front mirror to stop the electron beam. Tungsten was selected due to its short radiation length. The beam stop did not distort the mirror and the objective was realigned after installation. During the experiment X-rays 68 were generated along the beam line and were a significant source of noise. To minimize the number of X-rays reaching the camera (EMCCD chip) it was placed 12” below the beam path and surrounded with lead shield- ing (not shown). Sufficient OTR light was generated and collected with low noise during the experiment. Pictures of experimental setup Microscope Objective 1” Turning Mirror Electron Beam Path Tungsten Beam Stop Image Path/Out of Vacuum Figure 4.5: In-vacuum imaging system components: microscope objective and 1” turning mirror. Capillary exit, capillary not shown, is 25 mm up- stream (to the left) of microscope objective. The microscope objective and 1” turning mirror were vacuum com- patible, Fig. 4.5. All other optical components were in atmosphere, Figure 4.6. The setup of the imaging system had a measured depth of field of 40 80m. The alignment process included alignment of the beam line HeNe laser independently to the imaging system and the capillary. The alignment procedure is detailed in Appendix C. 69 Pictures of experimental setup EMCCD Camera 3” Turning Mirrors Image Path Camera Lens Vacuum Window Figure 4.6: Out of vacuum imaging system configuration: 3” turning mir- rors, camera lens and EMCCD camera. The 135 mm camera lens is not shown. Note that the camera is 12” below the beam path and that lead shielding (not shown) surrounded the camera. 4.2 Resolution and Magnication of Imaging System With the full experimental setup, except for the capillary, the magni- fication and the resolution of the imaging system were measured. On the rear of the capillary holder/assembly a USAF 1951 resolution test target was attached and imaged, Figure 4.7. Analysis of this image reveals a mag- nification of 10:9x. This is close to the expected value for a simple two lens magnification system with the magnification defined asM = f 1 =f 2 withf 1 70 andf 2 being the focal lengths of the two lenses. The two lenses in this sys- tem aref camera lens = 135 mm andf microscope = 13:35 mm, yieldingM = 10:1x. For the EMCCD camera, this magnification would allow for images on the capillary window of (13m=pixel)=10:1x = 1:3m=pixel and would allow for the detection of 7 pixels/filament (for 10m feature size) at the best resolution of the system. The resolution was defined to be the spatial frequency (line pairs/mm) where the Modulation Transfer Function, MTF, is 50% of its peak. The resolution was measured, from Fig. 4.7, to be group 6 element 6, which corresponds to a resolution of 8:8m=line pair or 4:4m/line. This measured optical resolution is sufficient to measure the smallest expected filament size in the experiment. Resolution Images ~960μ μ μ μm ~960μ μ μ μm Element 6 Group 6 Group 7 Figure 4.7: Longitudinal (direct) filament imaging for the imaging system setup on beam line 1, but with a USAF 1951 resolution test target placed at the location of the Si/Au window on the downstream side of the capillary. 71 4.3 Summary/Conclusion Detailed the ATF linear accelerator electron beam and beam line con- figuration Overview of the plasma source and density diagnostic Design of a direct filament imaging using OTR light Configuration and setup of the imaging system Performance of the imaging system were measured, resolution (4:4m/line) and magnification (10:9x), and is sufficient for the smallest expected feature size 72 Chapter 5 CFI Experimental Results In this chapter I will present and discuss results where the current filamentation instability is observed and studied in a laboratory environ- ment. The experiment was conducted at the BNL-ATF with its 60 MeV electron beam and a plasma capillary discharge. Chapter 4 summarized the primary components of the experiment, including the linear accelerator, plasma source and the design and setup of a direct imaging system with micron scale resolution. In the experiment multiple filaments are observed and imaged transversely at the plasma exit with optical transition radiation, OTR. By varying the plasma density the transition between single and multiple filaments is found to bek p r 2:2. Scaling of the transverse fila- ment size with the plasma skin depth is predicted in theory and observed over a range of plasma densities. Lowering the bunch charge, and thus bunch density, suppresses the instability. Fork p r 1 plasma focusing is observed and the focused beam size scales with the beam density. All of the experimental results are in good agreement with the predictions of theory and simulations. While parameters in the experiment differ from those in theory (infinite beam and plasma) simulations support the experimental 73 results. These results represent the first observation and characterization of CFI. 5.1 Previous Experimensts Kapentanakos in 1974 [27] conducted an experiment with a weakly relativistic beam, b 1, yet the results exhibit features and characteristics similar to CFI. This experiment used a 500 KeV hollow electron beam prop- agating in a uniform magnetic field through a 68 cm long plasma. The trans- verse beam profile was recorded by the damage to a Lucite disk placed in the beam path. Evolution of the instability was observed by moving the disk along the length of the plasma. The growth rate of the instability was reduced by increasing the magnetic field and reducing the beam current. Emittance of the beam was increased by placing a foil in the beam path re- sulting in increased filament size. Radiation emission was observed and recorded, however the filament sizes were not reported. The features and characteristics (beam breakup, growth rate and radiation) are similar to CFI but for a weakly relativistic, not well diagnosed beam propagating in a mag- netic field raise questions about the beam-plasma instability at work. Two recent experiments [24, 53] with laser-driven plasma-based ac- celerators observed CFI through imaging plasma density gradients which showed filamentation along the longitudinal plane of the electron beam. The goal of Tatarakis et al. [53] was to study laser produced electron 74 beams for the fast igniter - inertial confinement fusion concept [51], section 1.2.3. The laser system was focused onto an Al or Mylar target 50 175m thick with a gas jet directly behind the target. Electrons were generated in the target and propagated into and ionized the gas, the thick target stopped the laser. Shadowgraphy was used to image the plasma density gradients and filamentation of the electron beam was inferred, an unexpected result. The authors concluded that filamentation of the beam would be significant in ICF concepts and needed to be suppressed. Huntington et al. [24] set out to observe longitudinal filamentation from the CFI. An LWFA [52] scheme with self trapping of electrons was used with different length gas jets. Transverse interferometry was used to image plasma density gradients along the plasma and with a 5 mm long nozzle multiple hollow plasma structures were observed. Both recent studies used OSIRIS [17], a fully explicit PIC code, to study the instability with laser and plasma parameters from their experi- ments and found qualitative support with their results. None of these ex- periments studied the basic features of the instability including the transi- tion from single to multiple filaments (or plasma focusing to CFI) and scal- ing of the filament’s size with plasma density. Neither of these experiments directly showed filamentation of the beam itself. 75 5.2 Data Collection In our experiment there is independent control of the plasma den- sity, transverse beam size at the capillary entrance and bunch charge. One limitation in the experiment is that there is no diagnostic for measurement of the bunch length for long bunches and a length of z 5 ps is assumed. This estimate is due to the space charge effects in the rf gun and without ad- ditional compression the rf gun always produces a bunch current of 100A. The beam and plasma parameters are measurable and repeatable during a scan. The characteristic expansion length in vacuum of the electron beam from its waist due to its finite emittance is given by its beta function, 0x;y = 0 2 0x;y =" N . With the beam parameters of Table 3.1, o = 117, 0x;y = 50m and" Nx;y = 8 mm mrad , 0x;y > 3 cm and 0x;y L p ,L p = 2 cm. Thus, the transverse size of the bunch at the plasma exit for events without plasma can be taken to be the same as that at the capillary entrance and used as the input condition for events with plasma. The incoming bunch x and y profiles without plasma are essentially Gaussian and characterized by their RMS sizes 0x and 0y . Measurements show that in general the bunch trans- verse size 0x;y does not vary by more than 10% RMS from event to event and over a typical measurement period. The relative bunch charge is measured in the accelerating cavity of 76 the linear accelerator. During a typical scan the charge in the accelerating cavity varies by 1% rms. The relative bunch charge is calibrated with a Faraday cup upstream of the capillary. Data collection includes alternately recording transverse images of the bunch at the plasma/capillary exit with (filamented beam) and without plasma (incoming beam). The relative charge of the electron bunch in the accelerator and the discharge current and plasma light time evolution are recorded for every event. Two types of plasma density scans were used for data acquisition. Continuous scans, to study the evolution of the instability as a function of k p 0x;y , record one event at 55 different densities over the full plasma density range, from 10 13 to 7 10 17 cm 3 corresponding to plasma skin depths from 1680 to 6m. Discrete scans, to study the scaling of the filament size with plasma skin depth and to capture the variations due to the instability nature of the interaction. The discrete scan records ten events at six different plasma densities from from 1:6 10 16 to 3:3 10 17 cm 3 corresponding to plasma skin depths from 42 to 9m. For data analysis the occurrence of CFI is defined to be when mul- tiple filaments are observed with the plasma and when on average, the fil- ament size scales with the plasma skin depth. Note that reduction of the bunch transverse size at the plasma exit with k p 0x;y 1 is interpreted as plasma focusing by the underdense plasma [28], see section 2.2. 77 5.3 Experimental Results With the direct imaging system, section 4.1.3, which records trans- verse beam profiles at the capillary/plasma exit and with control of the re- peatable beam and plasma parameters several characteristics of the insta- bility can be studied. These include 1) observing the presence of multiple bunch filaments, 2) identifying the transition point, k p r , between plasma focusing (expected fork p r 1) and CFI (expected fork p r 1), 3) deter- mine scaling of the transverse filament size for the instability and plasma focusing and 4) control of the growth rate through suppression of the insta- bility. 5.3.1 Transverse Imaging of Beam and Multiple Filaments One representative transverse bunch image without plasma and five images with plasma and exhibiting focusing or filamentation are shown in Figure 5.1. In practice a filament is identified as a high count feature in the images. For these events the electron bunch parameters are 0x 80m and 0y 50m, chargeQ 1:0 nC and the plasma skin depth and values of k p 0x and k p 0y are 41:6m, 1:9, 1:2 for Fig. 5.1b, 15:4m, 5:2, 3:3 for Figs 5.1c and d and 12:3m, 6:5, 4:1 for Figs 5.1 e and f, respectively. The single feature in Fig. 5.1b is considered to be plasma focusing (single feature and k p r < 2:2) and Fig. 5.1c through f are CFI (multiple features, k p r > 2:2 and scaling of transverse filament size with 78 plasma skin depth). Figure 5.1 has b) 1, c) 5, d) 3, e) 3 and f) 4 filaments respectively. Note that in all cases, Figs 5.1 b) to f), the features identified as filaments have a higher peak count than the incoming bunch, Fig. 5.1 a), and therefore larger current densities. The total count is conserved for events with and without plasma to an average difference of 14%. These differences are attributed to incoming charge variations, fluctuations due to relatively low number of counts per pixel as well as the random additions of counts due to beam generated X-rays. The images in Fig. 5.1 illustrate the random character of the instability. The filaments size, position and number change with plasma density (see below). They also change from event to event, even with similar experimental conditions, a typical behavior for an instability-driven interaction. 5.3.2 Beam-Plasma Interaction Regime As the plasma density is decreased from values such thatk p 0x;y 1 where multiple filaments are present tok p 0x;y 1 only single “filament” events are expected. Figure 5.2 shows again the instability nature of the bunch filamentation through the number of filaments observed at various plasma densities, but displayed as a function ofk p 0y . To locate the transi- tion from single filament events, expected fork p 0y . 1, to multiple filament events, expected for k p 0y > 1, from a large data set both the continuous and discreet scan data are used. Additionally, 120 events were recorded at 79 Low to High Density 69 0 50 100 150 200 250 300 0 50 100 150 200 a) d) e) f) c) b) Plasma Off 180μ μ μ μm 128μ μ μ μm X Y Figure 5.1: OTR images of the bunch at the capillary exit with arrows indicating filaments. Bunch parameters are 0x 80m, 0y 50m and charge Q ' 1:0nC. a) without plasma and b) n e = 1:6 10 16 cm 3 (c=! pe = 41:6m), c) and d)n e = 1:2 10 17 cm 3 (c=! pe = 15:4m) and e) and f)n e = 1:9 10 17 cm 3 (c=! pe = 12:3m). X-rays are seen in the images and are a few pixels in size. Color tables are the same for b) through f). k p 0y 4:1 and showed between one and four filaments. Therefore single points in Fig. 5.2 in general correspond to multiple events with the same number of filaments. There is a very clear transition between single and multiple filament occurrences around k p 0y 2:2. Most noticeably there are no cases of multiple filaments fork p 0y < 2:2. Some instances of single filaments are present for k p 0y > 2:2, but the number of filaments can be as large as five. This variability is attributed to the sensitivity of the CFI to slight changes in plasma and bunch initial conditions. Fork p 0y > 4:5 only 80 one or two filaments are observed, which could be due to the merging of filaments [50]. Merging is also seen in simulations even though Eqn. 2.6 shows the growth rate is independent of plasma density. However, simula- tions in section 3.3.3 show that the growth rate is dependant on the plasma density and the saturation length is longer for higher plasma densities. Transition Radiation - Figure 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 Number Filaments k p σ σ σ σ oy Merging Focusing CFI Increasing n e Figure 5.2: Number of filaments observed in single events as a function of the CFI parameterk p 0y . For this measurement the chargeQ' 1:0 nC and 0x 80m, 0y 50m. Similar events lead to only single filaments for k p 0y < 2:2 and multiple (one to five) filaments fork p 0y > 2:2. Fork p 0y > 4:5 only one and two filaments are seen and could be due to merging of filaments. In general single points represent multiple events at the same k p 0y . 5.3.3 Transverse Filament Size Scaling The RMS transverse filament sizes is determined fromx andy projec- tions by selecting a small region around the features identified as filaments. 81 On a scan to scan basis the number of available (symmetric profiles)x andy filament projections for multiple filament events depends on the location of the filaments, see Fig. 5.1. The data set from a discrete scan that was used, was selected based on the largest number of available x or y projections for multiple filament events. For example, in Fig. 5.3 there are 29 and 13 availablex andy projections respectively and thex projections were used. Root mean squared filament sizes measured with the incoming bunch sizes 0x = 81m and 0y = 53m and chargeQ' 1:0 nC, named high charge, are shown in Figure 5.3. For this scan, multiple filaments, one to five, were observed for 9 c=! pe 20m corresponding to 2:7 < k p 0y < 5:8 on Fig- ure 5.2, that is over about one decade in plasma density. On average the filament size scales with the plasma skin depth for 12 c=! pe 42m. For the skin depth< 10m, the highest plasma density, the filaments size is larger thanc=! pe , which could be due to merging of the filaments. 5.3.4 Instability Growth as a Function of Beam Density The growth rate of the instability, Eq. 2.6, scales as p n b p Q=( 0x 0y 0z ), Eq. 3.5, and reducing the bunch charge reduces the development of CFI. The high charge scan in Figure 5.3 was repeated with half the charge, Q 0:54 nC, named the low charge, and approximately the same transverse bunch profile, 0x 89m and 0y 45m and the measured transverse bunch sizes are also shown in Figure 5.3. 82 73 increasing n e Low Charge (0.54nC) High Charge (1.0nC) Figure 5.3: Transverse filament size (RMS): red markers high charge scan - Q ' 1:0 nC and 0x = 81m, 0y = 53m. Blue markers low charge scan - Q' 0:54 nC and 0x ' 89m, 0y ' 45m. Both scans consist of six different plasma densities with each ten events recorded. The solid line represents exact correlation between filament size and plasma skin depth. For the high charge case we expect the bunch length to be longer than the low charge case due to space charge, yet we assume the same length in both cases. Evaluation of the growth rate given in Eq. 2.6 with the bunch parameters for the high and low charge case, shown in Table 5.1, indicate a growth of the CFI by 2:7 e-folding over the 2-cm long plasma and 2:1 e- folding for low charge bunch. This low charge scan generated only single “filaments” at all six plasma densities in contrast with the high charge scan where we observe one to five filaments for c=! pe 20m. In addition, Fig. 5.3 shows that while the higher charge events follow the skin depth scaling (except at the highest plasma density as mentioned above) the low 83 charge transverse sizes exhibit a completely different dependence on the skin depth. Note that 0y varies by a larger amount at lower charge and explains the larger spread in filament size, 17% RMS compared to 10% RMS for high charge. Parameters High Charge Low Charge Lorentz Factor - o 117 117 Relative Velocity - o 0.99996 0.99996 X Transverse Bunch Size - x (m) 53 45 Y Transverse Bunch Size - y (m) 81 89 Z Transverse Bunch Size - z (ps) 5 5 Bunch Charge -Q(nC) 1.00 0.54 Beam Particles -N b 6.2E+09 3.4E+09 Beam Density -n b (cm 3 ) 6.2E+13 3.6E+13 Growth Rate - (s 1 ) 4.1E+10 3.1E+10 Growth Rate - =c (cm 1 ) 1.4 1.0 Growth Length -c=(cm) 0.7 1.0 Number e-foldings -L p =(c=) 2.7 2.1 Table 5.1: Beam parameters, from discrete scan, for the high and low charge cases, Fig. 5.3, and corresponding growth rate, growth length and number of e-foldings overL p = 2 cm. Data from a continuous scan, Figure 5.4, taken at even lower plasma densities,c=! pe < 2; 000m, shows forc=! pe > 100m,k p 0y 1, that the dependencies of transverse bunch size on the plasma density are similar for the two charge cases with sizes consistently larger in the lower charge case. 84 Transition Radiation - Figure 0 50 100 150 0 500 1000 1500 2000 RMS Filament X-Size (μ μ μ μm) Plasma Skin Depth - c/ω ω ω ω p (μ μ μ μm) Increasing n e High Charge (1.0nC) Low Charge (0.54nC) Figure 5.4: Transverse filament size (RMS): red markers high charge scan - Q' 1:0 nC and 0x = 80m, 0y = 53m. Blue markers low charge scan - Q' 0:54 nC and 0x ' 91m, 0y ' 48m. Both scans consist of 56 different plasma densities each with one event recorded. The solid line represents exact correlation between filament size and plasma skin depth. This is consistent with the fact that plasma focusing remains, simply with a different reduction in transverse size and does not scale directly withc=! pe but with the bunch charge. All else equal, in the linear regimen b =n p 1, the transverse focusing force scales with the beam density (section 2.2) and smaller focused beam sizes are expected for the higher charge case at the same plasma density. 5.4 Chapter Conclusions Presented experimental evidence of CFI in a laboratory setting with an electron beam and plasma capillary discharge under the parameters 85 predicted in theory,k p r 1 and a relativistic beam. The results are based upon transverse imaging of the filaments at the plasma exit with OTR. The transition from multiple to single “filaments” (or focusing) was established, k p r 2:2, by varying the plasma density through the CFI parameterk p r . Observed scaling of the filament size with the plasma skin depth over a range of plasma densities. Showed that by slowing the growth rate of CFI, by reducing the bunch density, CFI was not observed, either through multiple filaments or scaling with the plasma density. At low plasma density but for high and low charge, transverse fea- ture sizes follow similar trends proportional to the bunch charge as expected in the plasma focusing regime. 86 Chapter 6 Conclusions 6.1 Summary The goal of this thesis was to present the first conclusive proof of the existence of CFI and to study basic characteristics of the instability through a well controlled experiment. In Chapter 1 motivation for such an experimen- tal study was presented and detailed three exciting active scientific research areas where CFI plays a role. In Chapter 2 the regime favorable to CFI was detailed and the evolution and main characteristics of the instability were laid out. Some of the key characteristics of CFI were observed in the exper- imental results and included breakup of the beam into high current density filaments, scaling of the growth rate of the instability with beam density, scaling of the transverse filament size with the plasma skin depth and the transition from plasma focusing to CFI. Chapter 3 presented simulation results showing CFI for beam and plasma parameters similar to those in the experiment. These results ver- ified key characteristics for CFI and showed that theoretical results for a beam and plasma of infinite extent hold for the finite beam and plasma of the experiment. Characteristics of the instability were studied by varying 87 beam (charge/density and emittance) and plasma (density) parameters. A dependence of the growth rate on plasma density, not predicted in theory, was shown to reduce the growth rate as the plasma density increased. These simulations also provided insight for the experimental results. Finally, one efficient method of externally seeding the instability was proposed. The experimental facility, design and setup were reviewed in Chap- ter 4. The design of the imaging system that allowed for direct imaging of the beam and filaments at the plasma exit was detailed and performance of the system was reported (resolution < 10m and magnification 10X) that was sufficient for the beam and plasma parameters in the experiment. This imaging system design will also be used in other experiments requir- ing direct beam imaging with high resolution. Experimental results were presented in Chapter 5 and are the first to conclusively show the basic char- acteristics of CFI and are published in a Physical Review Letter. These re- sults showed filamentation of the beam, the transition from CFI to plasma focusing by varying the plasma density, scaling of the transverse filament size with plasma skin depth, suppression of the instability by lowering the bunch charge and scaling of the focused transverse beam size with charge (beam density). 88 6.2 Future Work After this first study of CFI that confirmed the basic characteristics of the instability many exciting research opportunities are available to ex- tend understanding of the instability. These include studying the limit CFI places on PWFA applications, impact CFI has on energy transport in the fast ignitor-ICF concept, the radiation spectrum of CFI and comparison to the spectra of the afterglow of gamma ray bursts, further characteristics of the instability (growth rate, merging of filaments and magnetic field enhance- ment) and shocks driven by CFI. In the long bunch PWFA (see chapter 1) experiments are designed such that k p r 1 to avoid the CFI regime. This limits the maximum plasma density (k p n 1=2 e ) which also limits the maximum acceleration gradient achievable. Further experiments are needed to understand the up- per limit onk p r and the effect on the beam when approaching this limit. In the experiment detailed in chapter 5 the beam and plasma param- eters of the ATF were limited. For the fast ignitor-ICF concept a study with real beam and plasma parameters from an LWFA concept needs to be con- ducted to study the impact CFI has on energy deposition by the beam of hot electrons. The occurrence of CFI increases the divergence of the hot electron beam and reduces the density of the energy deposition. The presence of CFI will produce beam filaments, enhanced trans- 89 verse magnetic fields and plasma density gradients. Figure 6.1 shows a possible setup of three diagnostic tools that can run in parallel for each ex- perimental event (beam-plasma interaction). The direct filament imaging diagnostic was discussed in Chapter 4, but access to longitudinal diagnos- tics would allow for measuring the magnetic field enhancement (by Fara- day rotation), imaging the plasma density gradients from filamentation (by Schlieren or shadowgraphy) and merging of the filaments. Laser Beam Knife Edge Polarizer Vertical Capillary x z CCD Polarizer 45 o CCD Faraday Rotation Schlieren EMCCD OTR Direct Imaging Figure 6.1: Proposed transverse and longitudinal diagnostics setup. The plasma is assumed homogeneous and isotropic in absence of the electron beam, but when CFI is present plasma density gradients are ex- 90 pected since the high density beam filaments expel plasma electrons from their volume. The Schlieren method [21] can be used to image plasma den- sity gradients. The Schlieren diagnostic setup, see Fig. 6.1, uses a laser beam that passes transversely through the plasma-beam interaction region (transparent capillary). The light is then imaged onto a CCD camera. The presence of CFI will significantly increase the transverse mag- netic fields. The presence of these fields can be inferred by Faraday rotation measurements [25]. The Faraday rotation angle,', is: ' = o Z x n e (x)B x (x)dx (6.1) where B x is the component of the CFI generated magnetic field parallel to the propagation direction of the laser beam with wavelength o . The integration is over the transverse dimension of the plasma, along the laser propagation direction. Note that bothB x andn e vary along the integration path. The Faraday rotation diagnostic, see Fig. 6.1, can use the same laser beam as the Schieleren diagnostic. Deploying the Schlieren and Faraday rotation diagnostics requires an optically transparent material for the capillary body, such as sapphire, which is not currently available at the ATF. When CFI is present the bunch particles propagate in the signif- icantly increased magnetic fields and lead to generation of synchrotron (magnetic field is constant over one or more Larmor radius) or ”jitter” 91 (magnetic field changes over a Larmor radius) radiation. This radiation could be a signature of CFI in the afterglow of GRBs. The frequency of the radiation could be determined from simulations and from this the detection system (optics, detector, etc) can be designed. This diagnostic is not shown in Fig. 6.1 because the radiation will be intercepted by the direct imaging system (Si/Au window, microscope objective and turning mirror). Therefore, the radiation diagnostics could be performed at the same time as the Schlieren and Faraday rotation diagnostics but without direct filament imaging. Studying and comparing the growth rate to theory will require ac- curate measurement of the bunch charge and transverse and longitudinal size and either a short pulse laser or different length capillaries. In simu- lations and the experiment questions regarding dependence of the growth rate on plasma density were raised and need to be studied. Finally, in the experimental results, was merging of filaments at higher plasma densities (k p r > 4:5) the mechanism responsible for fewer observed filaments seen? Since the results presented here are the first conclusive ones it is likely that the experimental study will continue. It is interesting because it is a fundamental beam-plasma instability and because of its possible implications for advanced, plasma based accelerators, inertial confinement fusion and astrophysics. 92 Bibliography [1] “Collapsar model,” http://www.tapir.caltech.edu/ piro/research/ collapsar.html. 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Muggli, “A proposed demonstration of an experiment of protondriven plasma wakefield acceleration based on cern sps,”JournalofPlasmaPhysics, pp. 1–7, July 2012. 99 Appendix A Specication/Data sheet for Princeton Instru- ments EMCCD Camera - ProEM1024B IMAGING GROUP Applications: Single molecule detection, spectroscopy, chemiluminescence, astronomy, adaptive optics, hyperspectral imaging, phosphor imaging and tomography The ProEM+: 1024B eXcelon™3 and the ProEM+: 1024B EMCCD cameras from Princeton Instruments are the most advanced EMCCD cameras on the market, utilizing the latest low-noise readout electronics and a 1024 x 1024 EMCCD to deliver single photon sensitivity. These true 2-in-1 cameras feature a high speed EM mode to capture fast kinetics, a low speed normal CCD mode with very low read noise for precision photometry applications, and advanced features such as solid baseline stability and linear EM gain control. The ProEM+: 1024B series of EMCCD cameras are cooled to below -75° C using either air or liquid, or a combination of both, while the all metal, hermetic vacuum seals are warrantied for life – the only such guarantee in the industry. These cameras also feature, for the first time, the latest Gigabit Ethernet (GigE) interface to allow remote operation over a single cable without the need for custom framegrabbers. High QE and ultra low-noise electronics make the ProEM+: 1024B series of EMCCD cameras ideal for demanding, low-light level applications such as astronomy and Bose-Einstein Condensate (BEC) imaging. FEATURES BENEFITS eXcelon3 technology Higher QE in the UV and near IR regions; extremely low etaloning Electron multiplication (EM) gain Low-noise, impact-ionization process for single-photon sensitivity OptiCAL Linear, absolute EM gain calibration using built in precision light source; EM and Non-EM modes for the lowest noise and the best linearity BASE Baseline Active Stability Engine - stable reference for quantitative measurements PINS Princeton Instruments Noise Suppression technology. Independently optimized EM and non-EM modes for the lowest noise and the best linearity. Frame-transfer architecture Allows 100% duty cycle imaging for tracking applications Deep cooling Thermoelectric cooling below -75° C minimizes dark current and allows long exposure times; Camera can be cooled with air or liquid, or a combination of both, and fan can be permanently turned off for vibration-sensitive environments Single optical window Vacuum window is the only optical surface between incident light and the CCD surface; No losses due to multiple optical surfaces Built-in shutter Conveniently capture dark reference frames and protect camera from dust when not in use Dual amplifiers Individually optimized signal chains for a true 2-in-1 camera configuration, for high speed (EM mode) or long integration (normal CCD mode) applications 16-bit digitization Wide dynamic range to capture dim and bright signals in a single image 10 and 5 MHz readout Video rates at full-frame resolution. Use ROI/binning for hundreds of frames per second 100 kHz readout Noise performance of a slow scan camera for precise photometry applications Kinetics readout mode Powerful readout mode offers microsecond time resolution between sub-frames Available Mounts C-mount (standard), Canon mount and adjustable C-to-Spectroscope mount - easily attaches to microscopes, standard lenses, telescopes or other optical instruments Gigabit Ethernet (GigE) interface Industry standard for fast data transfer over long distances LightField™ for 64-bit Windows 7, or WinSpec for 32-bit Windows Flexible software packages for data acquisition, display and analysis; LightField offers intuitive, cutting edge user interface, IntelliCal™ and more PICAM* (64-bit) / PVCAM (32-bit) software development kits (SDKs) Compatible with Windows 7*/XP , Vista and Linux; Universal programming interfaces for easy custom programming LabVIEW™ Scientific Imaging Tool Kit (SITK ®) Pre-defined LabView vis provide easy integration of the camera into complex experiment setup ProEM+: 1024B Powered by LightField™ Page 1 of 5 Rev. P1 ProEM+: 1024B eXcelon3 ProEM+: 1024 shown with lens, sold separately. IMAGING GROUP ROI/Bin 1024 x 1024 512 x 512 256 x 256 128 x 128 1 x 1 8.9 17 33 62 2 x 2 17 33 62 108 4 x 4 33 61 106 168 8 x 8 61 104 162 225 NOTE: Frame rate measured at 10 MHz digitization and 800 nsec/row vertical shift. “Custom chip” mode increases frame rate at reduced ROI by 2x to 4x. * CTE and image quality are optimized even at the fastest vertical shift rate. FRAME RATE (fps) SPECIFICATIONS All specifications subject to change EM mode Normal CCD mode Read noise (typical) 20 e- rms @ 5 MHz 40 e- rms @ 10 MHz Read noise effectively reduced to <1 e- rms with on-chip multiplication gain enabled 2.5 - 3 e- rms @ 100 kHz 6 e- rms @ 1 MHz 10 e- rms @ 5 MHz Full well (typical) 730 ke- (output node) 80 ke- (single pixel) Non-Linearity <2% <1% Analog gain (typical) 12, 6, 3 e-/ADU 3, 1.5, 0.8 e-/ADU Deepest cooling temperature (@ +20° C ambient) __ 55° C +/- 0.05° C (guaranteed) Maximum Cooling: -65° C (air), -70° C (+20° C liquid), -75° C (+10° C liquid) Dark current @ _ 55° C 0.002 e-/p/sec (typical) 0.04 e-/p/sec (maximum) Clock induced charge (CIC) (typical) 0.01 e-/pixel/frame measured at ~1000x multiplication gain Electron multiplication (EM) gain 1 to 1000x, controlled in linear, absolute steps Digitization 16 bits @ 10 MHz, 5 MHz, 1 MHz and 100 kHz Vertical shift rate* 800 nsec/row - 5 μ sec/row (variable) Binning Flexible binning in vertical and 2x to 32x in horizontal Operating systems supported Windows XP/Vista/7 (32-bit), Windows 7 (64-bit) and Linux (64-bit) I/O signals Exposure, Readout, Trigger In, Image Shift, Waiting for Trigger Operating environment 0 to 30° C ambient, 0 to 80% relative humidity, non-condensing ProEM+:1024B eXcelon3 ProEM+: 1024B Features Back-illuminated CCD. High sensitivity and extremely low etaloning, grade 1, AIMO Back-illuminated CCD. High sensitivity in visible region CCD sensor Princeton Instruments proprietary CCD, grade 1, AIMO e2v CCD 201, grade 1, AIMO CCD format 1024 x 1024, 13 μ m imaging pixels 13.3 x 13.3 mm imaging area (optically centered) 1024 x 1024, 13 μ m imaging pixels 13.3 x 13.3 mm imaging area (optically centered) Shutter Electromechanical Mechanical Page 2 of 5 Rev. P1 ProEM+: 1024B eXcelon3 IMAGING GROUP Page 3 of 5 Rev. P1 ProEM+: 1024B eXcelon3 0 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600700 800 900 10001100 Back-Illuminated EMCCD eXcelon3 EMCCD Optional UV coating Quantum Eciency (%) Wavelength (nm) QUANTUM EFFICIENCY CURVE NOTE: Graph shows typical Quantum Efficiency (QE) data measured at + 25° C, representing expected performance at this temperature. Quantum Efficiency is a function of temperature and actual results will depend upon CCD temperature. IMAGING GROUP Page 4 of 5 Rev. P1 ProEM+: 1024B eXcelon3 NOTE: Standard anti-reflection (AR) coatings shown. Custom AR coatings and wedge window options are also available. Contact your local sales representative for more information. 50 55 60 65 70 75 80 85 90 95 100 200 300 400 500 600 700800 9001000 Transmission (%) Wavelength (nm) NIR-AR NO -AR MgF 2 BBAR (400 -1100nm) VACUUM WINDOW AR COATINGS Data taken with white light source through a monochromator comparing etaloning performance of eXcelon vs conventional back-illuminated EMCCDs. eXcelon PERFORMANCE IMAGING GROUP 0 0.24 6.20 1.08 27.41 2.59 65.89 3.14 79.86 2.08 52.73 2.66 67.46 0 2.35 59.7 2.90 73.7 2.90 73.7 2.35 59.7 3.89 98.7 4X #8-32 BUTTON HEAD HEX SCREWS .50 [12.5] 1.00-32 C-MOUNT ADJUSTABLE 4.063-20 UN MALE/FEMALE THREAD 10-32 UNF 0.313 0 0.690 CCD NOMINAL LOCATION 17.53 3.38 85.9 7.89 200.3 8.02 203.7 INLET/OUTLET COOLANT PORTS 3/8" QUICK DISCONNECT FITTINGS 0 0.62 15.7 1.73 43.8 0.63 15.9 1.76 44.6 0 0.24 6.0 1.49 37.7 0.50 12.8 1.25 31.9 STATUS LED 2-PIN LEMO MCX CONNECTORS 0 0 1.15 TRIPOD HOLES ALL 4 SIDES 29.2 0.90 22.9 1.13 28.6 1.13 28.6 NOTES: 25mm INTERNAL SHUTTER. 2. WINDOW MATERIAL FUSED SILICA THICKNESS .125" [3.18mm]. 3. UNIVERSAL TRIPOD ADAPTER SUPPLIED. 4. MCX -TO-BNC CABLE ADAPTERS SUPPLIED 5. POWER CONSUMPTION 80 WATTS. 6. (2) 10 FOOT TUBES AND MATING QUICK DISCONNECT LIQUID FITTINGS ARE SUPPLIED. 7. DIMENSIONS IN INCHES [MM]. 1. www.princetoninstruments.com info@princetoninstruments.com TOLL-FREE +1.877.474.2286 | PHONE +1 609.587.9797 Visit the Princeton Instruments website to find a dealer in your area. Page 5 of 5 Rev. P1 ProEM+: 1024B eXcelon3 ProEM+: 1024B OUTLINE DRAWING 105 Appendix B Specication/Data sheet for Ealing 25-0506-000 microscope objective X15 Specifications Numerical a per tur e . . . . . . . . . . . . . . . . . . . .0.28 . . . . . . .0.5 Magnifica tion (a t 160 mm tube length . . . . .X15 . . . . . . . X15 F ocal length (mm) . . . . . . . . . . . . . . . . . . . . 13.35 . . . . . 13.41 Visual field-of-vie w a t a bject (mm) . . . . . . . . . 1.2 . . . . . . .1.2 Obscura tion (%) . . . . . . . . . . . . . . . . . . . . . . .18.9 . . . . . .19.5 Appr o x. wor k ing distance (mm) . . . . . . . . . .24.5 . . . . . .23.2 Surf ace accurac y . . . . . . . . . . . . . . . . . . . . . . . λ /8 . . . . . . . λ /8 CAT AL OG NUMBER X15/coa ting Visible 25-0506-000 and 25-0555-000 Ultra Violet 25-0506-190 and 25-0555-190 25-0506-020 and 25-0555-020 W AVELENGTH (nm) 250-700 193-400 800-10,000 R E F L E C T I V I T Y % (a verage per surf ace) 89 (a verage) 89 98 CO ATING TYPE Al/MgF 2 UV Enhanced Alumin um Gold X15 X15 The X15 r ef lecting objecti ve is the lowest power Scharzschild objecti ve a vaila ble fr om Ealing . It has the largest exit pupil of the range and is ther ef or e par ticular l y useful in critical light g a thering a pplica tions; f or example , in the infrar ed wher e energ y le vels ma y be quite low . Many a pplica tions emplo y this objecti ve with incident illumina tion, or with self-lumi- nous objects if wor k is being done in a high tempera tur e envir onment. The ex ceptionall y long wor k ing distance of the X15 objecti ve mak es it par ticular l y valua ble f or wor k with furnaces or debar vessels. The r ela ti ve centra tion of the two mirr or s in this unit is pr e - cisel y adjusted bef or e lea ving the f actor y and, in normal use , should not r equir e any fur ther adjustment. Objecti ves should, howe ver , be check ed periodicall y f or centra tion and, if r eadjustment does become necessar y , it ma y be carried out using the hex wr enches pr ovided. Detailed instr uctions f or this pr ocedur e ar e gi ven in the handbook supplied with each r ef lecting objecti ve . Standar d X15 r ef lecting objecti ves ar e supplied adjusted f or use a t 160 mm tube length in conjunction with a standar d 0.17 mm thick cover glass. Ealing can pr ovide a f actor y cor- r ection f or any tube length between 80 mm and infinity , and any cover glass thickness up to 3 mm. Our technical staf f will be ha ppy to discuss any special r equir ements. Each standar d X15 r e f lecting objecti ve is supplied with inter- nal mirr or s coa ted with alumin um, with a magnesium f luo- ride over coa t. This combina tion is useful in most a pplica tions involving low power le vels. F or a pplica tions in which the objecti ve is r equir ed to withstand higher power sour ces and deli ver the highest per centage of input energ y , the alterna - ti ve coa tings shown in the ta ble ar e a vaila ble . X25 Specifications Magnifica tion (a t 160 mm tube length) . . . . . . . . . . . . . . X25 F ocal length (mm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.0 Visual field-of-vie w a t a bject (mm) . . . . . . . . . . . . . . . . . . 0.72 Numerical a per tur e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.4 Obscura tion (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16.7 Appr o x. wor k ing distance (mm) . . . . . . . . . . . . . . . . . . . .14.5 Surf ace accurac y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /8 X25 X25 The X25 r ef lecting objecti ve is the ne west member of this unique series of objecti ves. It was designed to fill the g a p between the X15 and X36 r ef lecting objecti ves. The X25 objecti ve of fer s a long wor k ing distance (14.5 mm) and a large N .A. (.40) in a high power objecti ve . This unit ma y be used in many of the a pplica tions curr entl y suited f or the X15 objecti ve wher e wor k ing distance and a large exit pupil ar e less critical. The objecti ve is based on a similar housing design tha t is used f or the X15 objecti ve . Thus any corr ections f or tube length or cover glass thickness must be perf ormed a t the f ac - tor y . The r ela ti ve centra tion of the two mirr or s in this unit is pr e - cisel y adjusted bef or e lea ving the f actor y and, in normal use , should not r equir e any fur ther adjustment. Objecti ves should, howe ver , be check ed periodicall y f or centra tion and, if r eadjustment does become necessar y , it ma y be carried out using the hex wr enches pr ovided. Detailed instr uctions f or this pr ocedur e ar e gi ven in the handbook supplied with each r e f lecting objecti ve . Standar d X25 r ef lecting objecti ves ar e s u pplied adjusted f or use a t 160 mm tube length in conjunction with a standar d 0.17 mm thick cover glass. Ealing can pr ovide a f actor y cor- r ection f or any tube length between 80 mm and infinity , and any cover glass thickness up to 3 mm. Our technical staf f will be ha ppy to discuss any special r equir ements. Each standar d X25 r e f lecting objecti ve is supplied with inter - nal mirr or s coa ted with alumin um, with a magnesium f luo- ride over coa t. This combina tion is useful in most a pplica tions involving low power le vels. F or a pplica tions in which the objecti ve is r equir ed to withstand higher power sour ces and deli ver the highest per centage of input energ y , the alterna- ti ve coa ting shown in the ta ble ar e a vaila ble . CAT AL OG NUMBER X25/coa ting Visible 25-0514-000 Ultra Violet 25-0514-190 25-0514-020 W AVELENGTH (nm) 250-700 193-400 800-10,000 R E F L E C T I V I T Y % (a verage per surf ace) 89 (a verage) 89 98 CO ATING TYPE Al/MgF 2 UV Enhanced Alumin um Gold 107 Appendix C CFI Alignment Process 1. Disable GPOP9 (interfere with turning mirror) 2. Remove downstream (and upstream?) mirror assemblies (in Compton chamber) 3. Focus PI camera with lens at infinity (use computer in control room) 4. Move computer/monitor into experimental hall for PI camera 5. Align beam line HeNe to end of BL#1 camera/iris 6. Take image of HeNe on GPOP8 with FrameGrabber (location of HeNe) and note location relative to fiducials 7. Setup and align local HeNe (into GPOP7) onto beam line HeNe a. Use lens at window entering GPOP7 to focus at middle of capillary b. GPOP8 and PI Camera for vector c. Setup iris’s to set location 8. Measure size of local HeNe a. Calibrate camera on framegrabber 9. Measure size of beam line HeNe a. Calibrate camera on framegrabber 10. Experiment imaging system (Vector for Imaging System is set to Beam LineHeNe) 108 a. Vector is camera (install an iris as well?) and target b. Install external optics c. Install internal optics d. Align to beam line HeNe e. Mark on PI camera beam line HeNe (use to image HeNe on target) 11. Align capillary to local HeNe(VectorforCapillaryissettoHeNe) a. Vector is front of electrode (off angle camera) and HeNe spot through the capillary either image outside chamber (off of turning mirror or on iris after chamber) b. Check in two positions and two angles that HeNe is located in mid- dle of capillary c. Set back reflection location c.i. Make sure that only have to raise capillary to see back reflec- tion c.ii. Mark retro reflection d. Mark HeNe locations on monitors (analog 8) locations 12. Setup camera for off angle, looks at front (upstream side) of capillary a. Manually change video system, number 13 13. Resolution and magnification measurements a. Remove capillary b. Insert microscope objective c. Image microscope objective to ensure front mirror is centered in back mirror d. Set camera lens at infinity location e. Use low profile translations stage forz-direction focusing f. Image HeNe spot on the target onto the point we marked on PI camera 109 g. Can we use imaging of resolution target from external to chamber with pellicle? h. Move target to groups 6/7 into HeNe path, focus and record im- ages for measuring resolution and magnification, in white light 14. Install light shielding a. Remove window on Compton chamber (remove earlier in the pro- cess?) 15. Install pellicle a. Lighting - lamp with variac b. See target and light in focus c. could use HeNe to locate light position, light vector 16. Install capillary with no Si/Au window 17. Image capillary exit (rear electrode) with PI camera and see that it correlates with the image from the front of the capillary 18. Check HeNe passes through capillary 19. Install Capillary for last time a. Install OTR window in electrode b. Align front of capillary c. Set back reflection to marked position d. Iterate until satisfied with alignment e. Make sure electrode wires are not close to objective or translations stage f. Make sure gas hoses are not in the way of turning mirror or HeNe g. Ensure e- beam can pass unobstructed with capillary installed (raise capillary) h. Bolt down capillary flange and readjust (checking imaging) before pumping down 110 i. Check focusing without small side window j. Install small side window

Abstract (if available)

Abstract Plasma instabilities can produce strong anisotropies, accelerate particles to high energies and generate large electric and magnetic fields. The Current Filamentation Instability, CFI, is of central importance for the propagation of relativistic electron beams in plasmas. CFI has potential relevance to astrophysics (afterglow of gamma ray bursts), inertial confinement fusion (energy transport in the fast-ignitor concept) and beam-driven plasma-based accelerators (placing an upper limit on the plasma density and accelerating gradient). ❧ The work in this dissertation combines a review of theory, particle-in-cell simulations and design and execution of an experiment that resulted in the first conclusive observation of the current filamentation instability. Current theory of CFI is reviewed and discussed. Simulations, with a particle-in-cell code, were conducted to validate theory for the finite size bunch and plasma used in the work and provide insight for the experiment. The design of a high-magnification imaging system to observe the result of the instability and the setup of the experiment are detailed. Finally, the experimental results are presented and discussed. ❧ The experiment was conducted at the Accelerator Test Facility at Brookhaven National Laboratory with the 60MeV e⁻ bunch and 2cm long plasma with variable density. The experiment included the systematic study and characterization of the instability as a function of the beam charge and plasma density, or ratio of bunch transverse size to plasma skin depth c/ɷ_pe or 1/k_pe. The transverse beam profile and size (σᵣ) is measured directly at the plasma exit using optical transition radiation from a thin gold coated silicon window. Experimental results show the transition from plasma focusing (k_pσᵣ << 1) to CFI (k_pσᵣ >> 1) is characterized by the appearance of multiple (1-5) beam filaments and the linear scaling of the transverse filaments size with the plasma skin depth. Suppression of the instability is demonstrated when lowering the growth rate of the instability by reducing the beam charge. The experimental results are in excellent agreement with theory and simulations.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Allen, Brian A. (author)

Core Title Experimental study of the current filamentation instability

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 11/19/2012

Defense Date 11/16/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag CFI,OAI-PMH Harvest,plasma instability,plasma physics,PWFA

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Muggli, Patric (committee chair), Gundersen, Martin A. (committee member), Prata, Aluizio, Jr. (committee member), Shakeshaft, Robin (committee member)

Creator Email brian_allen02@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-114282

Unique identifier UC11290619

Identifier usctheses-c3-114282 (legacy record id)

Legacy Identifier etd-AllenBrian-1300-0.pdf

Dmrecord 114282

Document Type Dissertation

Rights Allen, Brian A.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA